Theorem abrexex 3974

Description: Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in the class expression substituted for B, which can be thought of as B(x). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path abrexexlem2 3973, abrexexlem1 3972, fvresex 3971, resfunexg 3685, and funimaexg 3681. See also abrexex2 3985.

Hypothesis

Ref	Expression
abrexex.1	⊢ A ∈ V

Assertion

Ref	Expression
abrexex	⊢ {y∣∃x ∈ A y = B} ∈ V

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

Proof of Theorem abrexex

Step	Hyp	Ref	Expression
1		abrexex.1	. . 3 ⊢ A ∈ V
2		class2set 2808	. . 3 ⊢ {z ∈ B∣B ∈ V} ∈ V
3	1, 2	abrexexlem2 3973	. 2 ⊢ {y∣∃x ∈ A y = {z ∈ B∣B ∈ V}} ∈ V
4		visset 1859	. . . . . . 7 ⊢ y ∈ V
5		eleq1 1577	. . . . . . 7 ⊢ (y = B → (y ∈ V ↔ B ∈ V))
6	4, 5	mpbii 191	. . . . . 6 ⊢ (y = B → B ∈ V)
7		ax-1 4	. . . . . . . . 9 ⊢ (B ∈ V → (z ∈ B → B ∈ V))
8	7	r19.21aiv 1759	. . . . . . . 8 ⊢ (B ∈ V → ∀z ∈ B B ∈ V)
9		rabid2 1816	. . . . . . . 8 ⊢ (B = {z ∈ B∣B ∈ V} ↔ ∀z ∈ B B ∈ V)
10	8, 9	sylibr 198	. . . . . . 7 ⊢ (B ∈ V → B = {z ∈ B∣B ∈ V})
11	10	eqeq2d 1529	. . . . . 6 ⊢ (B ∈ V → (y = B ↔ y = {z ∈ B∣B ∈ V}))
12	6, 11	syl 10	. . . . 5 ⊢ (y = B → (y = B ↔ y = {z ∈ B∣B ∈ V}))
13	12	ibi 595	. . . 4 ⊢ (y = B → y = {z ∈ B∣B ∈ V})
14	13	r19.22si 1780	. . 3 ⊢ (∃x ∈ A y = B → ∃x ∈ A y = {z ∈ B∣B ∈ V})
15	14	ss2abi 2172	. 2 ⊢ {y∣∃x ∈ A y = B} ⊆ {y∣∃x ∈ A y = {z ∈ B∣B ∈ V}}
16	3, 15	ssexi 2794	1 ⊢ {y∣∃x ∈ A y = B} ∈ V

Syntax hints:   ↔ wb 144   = wceq 992   ∈ wcel 994  {cab 1505  ∀wral 1691  ∃wrex 1692  {crab 1694  Vcvv 1857

This theorem is referenced by:  abrexexg 3975  oprvex 4102  iunon 4207  aceq5lem4 4884  aceq6b 4888  kmlem10 4920  fictblem 11422  fictb 11423  hartog 11436  compsublem 11487  compsub 11488  hscptsscld 11491  cncomp 11494  txbas 11973

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fv 3279

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