Theorem bnj1366 30900

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1366.1	|- ( ps <-> ( A e. _V /\ A. x e. A E! y ph /\ B = { y | E. x e. A ph } ) )

Assertion

Ref	Expression
bnj1366	|- ( ps -> B e. _V )

Distinct variable group:   x, A, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   B( x, y)

Proof of Theorem bnj1366

Step	Hyp	Ref	Expression
1		bnj1366.1	. . . 4 |- ( ps <-> ( A e. _V /\ A. x e. A E! y ph /\ B = { y | E. x e. A ph } ) )
2	1	simp3bi 1078	. . 3 |- ( ps -> B = { y | E. x e. A ph } )
3	1	simp2bi 1077	. . . . 5 |- ( ps -> A. x e. A E! y ph )
4		nfcv 2764	. . . . . . 7 |- F/_ y A
5		nfeu1 2480	. . . . . . 7 |- F/ y E! y ph
6	4, 5	nfral 2945	. . . . . 6 |- F/ y A. x e. A E! y ph
7		nfra1 2941	. . . . . . . 8 |- F/ x A. x e. A E! y ph
8		rspa 2930	. . . . . . . . 9 |- ( ( A. x e. A E! y ph /\ x e. A ) -> E! y ph )
9		iota1 5865	. . . . . . . . . 10 |- ( E! y ph -> ( ph <-> ( iota y ph ) = y ) )
10		eqcom 2629	. . . . . . . . . 10 |- ( ( iota y ph ) = y <-> y = ( iota y ph ) )
11	9, 10	syl6bb 276	. . . . . . . . 9 |- ( E! y ph -> ( ph <-> y = ( iota y ph ) ) )
12	8, 11	syl 17	. . . . . . . 8 |- ( ( A. x e. A E! y ph /\ x e. A ) -> ( ph <-> y = ( iota y ph ) ) )
13	7, 12	rexbida 3047	. . . . . . 7 |- ( A. x e. A E! y ph -> ( E. x e. A ph <-> E. x e. A y = ( iota y ph ) ) )
14		abid 2610	. . . . . . 7 |- ( y e. { y | E. x e. A ph } <-> E. x e. A ph )
15		eqid 2622	. . . . . . . 8 |- ( x e. A |-> ( iota y ph ) ) = ( x e. A |-> ( iota y ph ) )
16		iotaex 5868	. . . . . . . 8 |- ( iota y ph ) e. _V
17	15, 16	elrnmpti 5376	. . . . . . 7 |- ( y e. ran ( x e. A |-> ( iota y ph ) ) <-> E. x e. A y = ( iota y ph ) )
18	13, 14, 17	3bitr4g 303	. . . . . 6 |- ( A. x e. A E! y ph -> ( y e. { y | E. x e. A ph } <-> y e. ran ( x e. A |-> ( iota y ph ) ) ) )
19	6, 18	alrimi 2082	. . . . 5 |- ( A. x e. A E! y ph -> A. y ( y e. { y | E. x e. A ph } <-> y e. ran ( x e. A |-> ( iota y ph ) ) ) )
20	3, 19	syl 17	. . . 4 |- ( ps -> A. y ( y e. { y | E. x e. A ph } <-> y e. ran ( x e. A |-> ( iota y ph ) ) ) )
21		nfab1 2766	. . . . 5 |- F/_ y { y | E. x e. A ph }
22		nfiota1 5853	. . . . . . 7 |- F/_ y ( iota y ph )
23	4, 22	nfmpt 4746	. . . . . 6 |- F/_ y ( x e. A |-> ( iota y ph ) )
24	23	nfrn 5368	. . . . 5 |- F/_ y ran ( x e. A |-> ( iota y ph ) )
25	21, 24	cleqf 2790	. . . 4 |- ( { y | E. x e. A ph } = ran ( x e. A |-> ( iota y ph ) ) <-> A. y ( y e. { y | E. x e. A ph } <-> y e. ran ( x e. A |-> ( iota y ph ) ) ) )
26	20, 25	sylibr 224	. . 3 |- ( ps -> { y | E. x e. A ph } = ran ( x e. A |-> ( iota y ph ) ) )
27	2, 26	eqtrd 2656	. 2 |- ( ps -> B = ran ( x e. A |-> ( iota y ph ) ) )
28	1	simp1bi 1076	. . 3 |- ( ps -> A e. _V )
29		mptexg 6484	. . 3 |- ( A e. _V -> ( x e. A |-> ( iota y ph ) ) e. _V )
30		rnexg 7098	. . 3 |- ( ( x e. A |-> ( iota y ph ) ) e. _V -> ran ( x e. A |-> ( iota y ph ) ) e. _V )
31	28, 29, 30	3syl 18	. 2 |- ( ps -> ran ( x e. A |-> ( iota y ph ) ) e. _V )
32	27, 31	eqeltrd 2701	1 |- ( ps -> B e. _V )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   E!weu 2470   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   |-> cmpt 4729   ran crn 5115   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  bnj1489  31124