Theorem rabfmpunirn 29453

Description: Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)

Hypotheses

Ref	Expression
rabfmpunirn.1	|- F = ( x e. V |-> { y e. W | ph } )
rabfmpunirn.2	|- W e. _V
rabfmpunirn.3	|- ( y = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
rabfmpunirn	|- ( B e. U. ran F <-> E. x e. V ( B e. W /\ ps ) )

Distinct variable groups:   x, y, B   x, F, y   x, V, y   y, W   ps, y

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

Proof of Theorem rabfmpunirn

Step	Hyp	Ref	Expression
1		rabfmpunirn.1	. . . 4 |- F = ( x e. V |-> { y e. W | ph } )
2		df-rab 2921	. . . . 5 |- { y e. W | ph } = { y | ( y e. W /\ ph ) }
3	2	mpteq2i 4741	. . . 4 |- ( x e. V |-> { y e. W | ph } ) = ( x e. V |-> { y | ( y e. W /\ ph ) } )
4	1, 3	eqtri 2644	. . 3 |- F = ( x e. V |-> { y | ( y e. W /\ ph ) } )
5		rabfmpunirn.2	. . . 4 |- W e. _V
6	5	zfausab 4811	. . 3 |- { y | ( y e. W /\ ph ) } e. _V
7		eleq1 2689	. . . 4 |- ( y = B -> ( y e. W <-> B e. W ) )
8		rabfmpunirn.3	. . . 4 |- ( y = B -> ( ph <-> ps ) )
9	7, 8	anbi12d 747	. . 3 |- ( y = B -> ( ( y e. W /\ ph ) <-> ( B e. W /\ ps ) ) )
10	4, 6, 9	abfmpunirn 29452	. 2 |- ( B e. U. ran F <-> ( B e. _V /\ E. x e. V ( B e. W /\ ps ) ) )
11		elex 3212	. . . . 5 |- ( B e. W -> B e. _V )
12	11	adantr 481	. . . 4 |- ( ( B e. W /\ ps ) -> B e. _V )
13	12	rexlimivw 3029	. . 3 |- ( E. x e. V ( B e. W /\ ps ) -> B e. _V )
14	13	pm4.71ri 665	. 2 |- ( E. x e. V ( B e. W /\ ps ) <-> ( B e. _V /\ E. x e. V ( B e. W /\ ps ) ) )
15	10, 14	bitr4i 267	1 |- ( B e. U. ran F <-> E. x e. V ( B e. W /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   _Vcvv 3200   U.cuni 4436   |-> cmpt 4729   ran crn 5115

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-ral 2917  df-rex 2918  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-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-fv 5896

This theorem is referenced by: (None)