Theorem unima 39346

Description: Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
unima	|- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( F " ( B u. C ) ) = ( ( F " B ) u. ( F " C ) ) )

Proof of Theorem unima

Dummy variables y x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . 6 |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> F Fn A )
2		simpl 473	. . . . . . . 8 |- ( ( B C_ A /\ C C_ A ) -> B C_ A )
3		simpr 477	. . . . . . . 8 |- ( ( B C_ A /\ C C_ A ) -> C C_ A )
4	2, 3	unssd 3789	. . . . . . 7 |- ( ( B C_ A /\ C C_ A ) -> ( B u. C ) C_ A )
5	4	3adant1 1079	. . . . . 6 |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( B u. C ) C_ A )
6		fvelimab 6253	. . . . . 6 |- ( ( F Fn A /\ ( B u. C ) C_ A ) -> ( y e. ( F " ( B u. C ) ) <-> E. x e. ( B u. C ) ( F ` x ) = y ) )
7	1, 5, 6	syl2anc 693	. . . . 5 |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( y e. ( F " ( B u. C ) ) <-> E. x e. ( B u. C ) ( F ` x ) = y ) )
8		rexun 3793	. . . . 5 |- ( E. x e. ( B u. C ) ( F ` x ) = y <-> ( E. x e. B ( F ` x ) = y \/ E. x e. C ( F ` x ) = y ) )
9	7, 8	syl6bb 276	. . . 4 |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( y e. ( F " ( B u. C ) ) <-> ( E. x e. B ( F ` x ) = y \/ E. x e. C ( F ` x ) = y ) ) )
10		fvelimab 6253	. . . . . 6 |- ( ( F Fn A /\ B C_ A ) -> ( y e. ( F " B ) <-> E. x e. B ( F ` x ) = y ) )
11	10	3adant3 1081	. . . . 5 |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( y e. ( F " B ) <-> E. x e. B ( F ` x ) = y ) )
12		fvelimab 6253	. . . . . 6 |- ( ( F Fn A /\ C C_ A ) -> ( y e. ( F " C ) <-> E. x e. C ( F ` x ) = y ) )
13	12	3adant2 1080	. . . . 5 |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( y e. ( F " C ) <-> E. x e. C ( F ` x ) = y ) )
14	11, 13	orbi12d 746	. . . 4 |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( ( y e. ( F " B ) \/ y e. ( F " C ) ) <-> ( E. x e. B ( F ` x ) = y \/ E. x e. C ( F ` x ) = y ) ) )
15	9, 14	bitr4d 271	. . 3 |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( y e. ( F " ( B u. C ) ) <-> ( y e. ( F " B ) \/ y e. ( F " C ) ) ) )
16		elun 3753	. . 3 |- ( y e. ( ( F " B ) u. ( F " C ) ) <-> ( y e. ( F " B ) \/ y e. ( F " C ) ) )
17	15, 16	syl6bbr 278	. 2 |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( y e. ( F " ( B u. C ) ) <-> y e. ( ( F " B ) u. ( F " C ) ) ) )
18	17	eqrdv 2620	1 |- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( F " ( B u. C ) ) = ( ( F " B ) u. ( F " C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   u. cun 3572   C_ wss 3574   "cima 5117   Fn wfn 5883   ` cfv 5888

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-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-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:  icccncfext  40100