Theorem fvun2 5585

Description: The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Assertion

Ref	Expression
fvun2	|- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( F u. G ) ` X ) = ( G ` X ) )

Proof of Theorem fvun2

Step	Hyp	Ref	Expression
1		uncom 3281	. . 3 |- ( F u. G ) = ( G u. F )
2	1	fveq1i 5518	. 2 |- ( ( F u. G ) ` X ) = ( ( G u. F ) ` X )
3		incom 3329	. . . . . 6 |- ( A i^i B ) = ( B i^i A )
4	3	eqeq1i 2185	. . . . 5 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
5	4	anbi1i 458	. . . 4 |- ( ( ( A i^i B ) = (/) /\ X e. B ) <-> ( ( B i^i A ) = (/) /\ X e. B ) )
6		fvun1 5584	. . . 4 |- ( ( G Fn B /\ F Fn A /\ ( ( B i^i A ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
7	5, 6	syl3an3b 1276	. . 3 |- ( ( G Fn B /\ F Fn A /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
8	7	3com12 1207	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
9	2, 8	eqtrid 2222	1 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( F u. G ) ` X ) = ( G ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   u. cun 3129   i^i cin 3130   (/)c0 3424   Fn wfn 5213   ` cfv 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  caseinr  7093