Theorem fvun2 5770

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 3373	. . 3 |- ( F u. G ) = ( G u. F )
2	1	fveq1i 5696	. 2 |- ( ( F u. G ) ` X ) = ( ( G u. F ) ` X )
3		incom 3421	. . . . . 6 |- ( A i^i B ) = ( B i^i A )
4	3	eqeq1i 2246	. . . . 5 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
5	4	anbi1i 462	. . . 4 |- ( ( ( A i^i B ) = (/) /\ X e. B ) <-> ( ( B i^i A ) = (/) /\ X e. B ) )
6		fvun1 5769	. . . 4 |- ( ( G Fn B /\ F Fn A /\ ( ( B i^i A ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
7	5, 6	syl3an3b 1316	. . 3 |- ( ( G Fn B /\ F Fn A /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
8	7	3com12 1238	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
9	2, 8	eqtrid 2283	1 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( F u. G ) ` X ) = ( G ` X ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   u. cun 3218   i^i cin 3219   (/)c0 3520   Fn wfn 5372   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by:  fvun2d  5772  caseinr  7432  gsump1  14157  vtxdfifiun  16538