Theorem fvun2 5553

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 3266	. . 3 |- ( F u. G ) = ( G u. F )
2	1	fveq1i 5487	. 2 |- ( ( F u. G ) ` X ) = ( ( G u. F ) ` X )
3		incom 3314	. . . . . 6 |- ( A i^i B ) = ( B i^i A )
4	3	eqeq1i 2173	. . . . 5 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
5	4	anbi1i 454	. . . 4 |- ( ( ( A i^i B ) = (/) /\ X e. B ) <-> ( ( B i^i A ) = (/) /\ X e. B ) )
6		fvun1 5552	. . . 4 |- ( ( G Fn B /\ F Fn A /\ ( ( B i^i A ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
7	5, 6	syl3an3b 1266	. . 3 |- ( ( G Fn B /\ F Fn A /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
8	7	3com12 1197	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
9	2, 8	syl5eq 2211	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 103   /\ w3a 968   = wceq 1343   e. wcel 2136   u. cun 3114   i^i cin 3115   (/)c0 3409   Fn wfn 5183   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by:  caseinr  7057