Theorem funfvop 3810

Description: Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41.

Assertion

Ref	Expression
funfvop	|- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)

Proof of Theorem funfvop

Step	Hyp	Ref	Expression
1		fvex 3739	. . 3 |- (F` A) e. V
2	1	isseti 1818	. 2 |- E.x x = (F` A)
3		visset 1816	. . . . . . 7 |- x e. V
4	3	funopfvb 3763	. . . . . 6 |- ((Fun F /\ A e. dom F) -> ((F` A) = x <-> <.A, x>. e. F))
5		opeq2 2493	. . . . . . . 8 |- ((F` A) = x -> <.A, (F` A)>. = <.A, x>.)
6	5	eleq1d 1543	. . . . . . 7 |- ((F` A) = x -> (<.A, (F` A)>. e. F <-> <.A, x>. e. F))
7	6	biimprcd 156	. . . . . 6 |- (<.A, x>. e. F -> ((F` A) = x -> <.A, (F` A)>. e. F))
8	4, 7	syl6bi 214	. . . . 5 |- ((Fun F /\ A e. dom F) -> ((F` A) = x -> ((F` A) = x -> <.A, (F` A)>. e. F)))
9	8	pm2.43d 65	. . . 4 |- ((Fun F /\ A e. dom F) -> ((F` A) = x -> <.A, (F` A)>. e. F))
10		eqcom 1480	. . . 4 |- (x = (F` A) <-> (F` A) = x)
11	9, 10	syl5ib 206	. . 3 |- ((Fun F /\ A e. dom F) -> (x = (F` A) -> <.A, (F` A)>. e. F))
12	11	19.23adv 1216	. 2 |- ((Fun F /\ A e. dom F) -> (E.x x = (F` A) -> <.A, (F` A)>. e. F))
13	2, 12	mpi 44	1 |- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  <.cop 2416  dom cdm 3177  Fun wfun 3183  ` cfv 3189

This theorem is referenced by:  fvimacnv 3812  fnopfv 3818  fvelrn 3819  dff2 3824  funfvima3 3861  fundmen 4435  adjt 9859

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-pow 2749  ax-pr 2786  ax-un 2873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-id 2842  df-xp 3191  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-fv 3205

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