Theorem abrexex 2912

Description: Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in B. This simple-looking theorem is actually quite powerful and involves the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path abrexexlem2 2911, abrexexlem1 2910, fvresex 2909, resfunexg 2717, and funimaexg 2715. See also abrexex2 2915.

Hypothesis

Ref	Expression
abrexex.1	⊢ A ∈ V

Assertion

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

Distinct variable group(s):   x,y,A   y,B

Proof of Theorem abrexex

Step	Hyp	Ref	Expression
1		abrexex.1	. . 3 ⊢ A ∈ V
2		class2set 1747	. . 3 ⊢ {z ∈ B∣B ∈ V} ∈ V
3	1, 2	abrexexlem2 2911	. 2 ⊢ {y∣∃x ∈ A y = {z ∈ B∣B ∈ V}} ∈ V
4		visset 1350	. . . . . . 7 ⊢ y ∈ V
5		eleq1 1149	. . . . . . 7 ⊢ (y = B → (y ∈ V ↔ B ∈ V))
6	4, 5	mpbii 168	. . . . . 6 ⊢ (y = B → B ∈ V)
7		ax-1 3	. . . . . . . . 9 ⊢ (B ∈ V → (z ∈ B → B ∈ V))
8	7	r19.21aiv 1259	. . . . . . . 8 ⊢ (B ∈ V → ∀z ∈ B B ∈ V)
9		rabid2 1309	. . . . . . . 8 ⊢ (B = {z ∈ B∣B ∈ V} ↔ ∀z ∈ B B ∈ V)
10	8, 9	sylibr 175	. . . . . . 7 ⊢ (B ∈ V → B = {z ∈ B∣B ∈ V})
11	10	cleq2d 1112	. . . . . 6 ⊢ (B ∈ V → (y = B ↔ y = {z ∈ B∣B ∈ V}))
12	6, 11	syl 12	. . . . 5 ⊢ (y = B → (y = B ↔ y = {z ∈ B∣B ∈ V}))
13	12	ibi 449	. . . 4 ⊢ (y = B → y = {z ∈ B∣B ∈ V})
14	13	r19.22si 1275	. . 3 ⊢ (∃x ∈ A y = B → ∃x ∈ A y = {z ∈ B∣B ∈ V})
15	14	ss2abi 1552	. 2 ⊢ {y∣∃x ∈ A y = B} ⊆ {y∣∃x ∈ A y = {z ∈ B∣B ∈ V}}
16	3, 15	ssexi 1701	1 ⊢ {y∣∃x ∈ A y = B} ∈ V

Syntax hints:   ↔ wb 127  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348

This theorem is referenced by:  abrexexg 2913  iunex 2914  iunon 2947  oprvalex 3055  aceq5lem4 3561  aceq6b 3565  kmlem9 3588

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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