Theorem rnssopab 3841

Description: Range of a function that is expressed as an ordered-pair class abstraction.

Hypotheses

Ref	Expression
fopab2.1	|- F = {<.x, y>. | (x e. A /\ y = C)}
rnssopab.2	|- C e. V

Assertion

Ref	Expression
rnssopab	|- (A.x e. A C e. B <-> ran F (_ B)

Distinct variable groups:   x,y,A   x,B,y   y,C

Proof of Theorem rnssopab

Step	Hyp	Ref	Expression
1		fopab2.1	. . . 4 |- F = {<.x, y>. | (x e. A /\ y = C)}
2	1	fopab2 3839	. . 3 |- (A.x e. A C e. B <-> F:A-->B)
3		frn 3649	. . 3 |- (F:A-->B -> ran F (_ B)
4	2, 3	sylbi 199	. 2 |- (A.x e. A C e. B -> ran F (_ B)
5		hbopab1 2829	. . . . . 6 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.x z e. {<.x, y>. | (x e. A /\ y = C)})
6	1, 5	hbxfr 1570	. . . . 5 |- (z e. F -> A.x z e. F)
7	6	hbrn 3367	. . . 4 |- (z e. ran F -> A.x z e. ran F)
8		ax-17 975	. . . 4 |- (z e. B -> A.x z e. B)
9	7, 8	hbss 2073	. . 3 |- (ran F (_ B -> A.xran F (_ B)
10		ssel 2074	. . . 4 |- (ran F (_ B -> (C e. ran F -> C e. B))
11		rnssopab.2	. . . . . . 7 |- C e. V
12		fvopab2 3807	. . . . . . 7 |- ((x e. A /\ C e. V) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
13	11, 12	mpan2 700	. . . . . 6 |- (x e. A -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
14	1	fveq1i 3741	. . . . . 6 |- (F` x) = ({<.x, y>. | (x e. A /\ y = C)}` x)
15	13, 14	syl5eq 1526	. . . . 5 |- (x e. A -> (F` x) = C)
16	11, 1	fnopab2 3634	. . . . . 6 |- F Fn A
17		fnfvelrn 3829	. . . . . 6 |- ((F Fn A /\ x e. A) -> (F` x) e. ran F)
18	16, 17	mpan 699	. . . . 5 |- (x e. A -> (F` x) e. ran F)
19	15, 18	eqeltrrd 1556	. . . 4 |- (x e. A -> C e. ran F)
20	10, 19	syl5 21	. . 3 |- (ran F (_ B -> (x e. A -> C e. B))
21	9, 20	r19.21ai 1719	. 2 |- (ran F (_ B -> A.x e. A C e. B)
22	4, 21	impbii 157	1 |- (A.x e. A C e. B <-> ran F (_ B)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 960   e. wcel 962  A.wral 1652  Vcvv 1818   (_ wss 2058  {copab 2681  ran crn 3187   Fn wfn 3193  -->wf 3194  ` cfv 3198

This theorem is referenced by:  fopab3 3842  oprcn 8003  ip1cnilem2 8399  ip1cnilem3 8400  ipasslem6 8520  kbass2 10074

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-sep 2718  ax-pow 2758  ax-pr 2795  ax-un 2882

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-uni 2518  df-br 2635  df-opab 2682  df-id 2851  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fn 3209  df-f 3210  df-fv 3214

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