Theorem fvun2 5625

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 3304	. . 3 |- ( F u. G ) = ( G u. F )
2	1	fveq1i 5556	. 2 |- ( ( F u. G ) ` X ) = ( ( G u. F ) ` X )
3		incom 3352	. . . . . 6 |- ( A i^i B ) = ( B i^i A )
4	3	eqeq1i 2201	. . . . 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 5624	. . . 4 |- ( ( G Fn B /\ F Fn A /\ ( ( B i^i A ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
7	5, 6	syl3an3b 1287	. . 3 |- ( ( G Fn B /\ F Fn A /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
8	7	3com12 1209	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( G u. F ) ` X ) = ( G ` X ) )
9	2, 8	eqtrid 2238	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 980   = wceq 1364   e. wcel 2164   u. cun 3152   i^i cin 3153   (/)c0 3447   Fn wfn 5250   ` cfv 5255

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by:  caseinr  7153