Theorem funimaeq 39461

Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
funimaeq.x	|- F/ x ph
funimaeq.f	|- ( ph -> Fun F )
funimaeq.g	|- ( ph -> Fun G )
funimaeq.a	|- ( ph -> A C_ dom F )
funimaeq.d	|- ( ph -> A C_ dom G )
funimaeq.e	|- ( ( ph /\ x e. A ) -> ( F ` x ) = ( G ` x ) )

Assertion

Ref	Expression
funimaeq	|- ( ph -> ( F " A ) = ( G " A ) )

Distinct variable groups:   x, A   x, F   x, G

Allowed substitution hint:   ph( x)

Proof of Theorem funimaeq

Step	Hyp	Ref	Expression
1		funimaeq.x	. . . 4 |- F/ x ph
2		funimaeq.e	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( G ` x ) )
3		funimaeq.g	. . . . . . . 8 |- ( ph -> Fun G )
4	3	funfnd 5919	. . . . . . 7 |- ( ph -> G Fn dom G )
5	4	adantr 481	. . . . . 6 |- ( ( ph /\ x e. A ) -> G Fn dom G )
6		funimaeq.d	. . . . . . 7 |- ( ph -> A C_ dom G )
7	6	adantr 481	. . . . . 6 |- ( ( ph /\ x e. A ) -> A C_ dom G )
8		simpr 477	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. A )
9		fnfvima 6496	. . . . . 6 |- ( ( G Fn dom G /\ A C_ dom G /\ x e. A ) -> ( G ` x ) e. ( G " A ) )
10	5, 7, 8, 9	syl3anc 1326	. . . . 5 |- ( ( ph /\ x e. A ) -> ( G ` x ) e. ( G " A ) )
11	2, 10	eqeltrd 2701	. . . 4 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. ( G " A ) )
12	1, 11	ralrimia 39315	. . 3 |- ( ph -> A. x e. A ( F ` x ) e. ( G " A ) )
13		funimaeq.f	. . . 4 |- ( ph -> Fun F )
14		funimaeq.a	. . . 4 |- ( ph -> A C_ dom F )
15		funimass4 6247	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ ( G " A ) <-> A. x e. A ( F ` x ) e. ( G " A ) ) )
16	13, 14, 15	syl2anc 693	. . 3 |- ( ph -> ( ( F " A ) C_ ( G " A ) <-> A. x e. A ( F ` x ) e. ( G " A ) ) )
17	12, 16	mpbird 247	. 2 |- ( ph -> ( F " A ) C_ ( G " A ) )
18	2	eqcomd 2628	. . . . 5 |- ( ( ph /\ x e. A ) -> ( G ` x ) = ( F ` x ) )
19	13	funfnd 5919	. . . . . . 7 |- ( ph -> F Fn dom F )
20	19	adantr 481	. . . . . 6 |- ( ( ph /\ x e. A ) -> F Fn dom F )
21	14	adantr 481	. . . . . 6 |- ( ( ph /\ x e. A ) -> A C_ dom F )
22		fnfvima 6496	. . . . . 6 |- ( ( F Fn dom F /\ A C_ dom F /\ x e. A ) -> ( F ` x ) e. ( F " A ) )
23	20, 21, 8, 22	syl3anc 1326	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. ( F " A ) )
24	18, 23	eqeltrd 2701	. . . 4 |- ( ( ph /\ x e. A ) -> ( G ` x ) e. ( F " A ) )
25	1, 24	ralrimia 39315	. . 3 |- ( ph -> A. x e. A ( G ` x ) e. ( F " A ) )
26		funimass4 6247	. . . 4 |- ( ( Fun G /\ A C_ dom G ) -> ( ( G " A ) C_ ( F " A ) <-> A. x e. A ( G ` x ) e. ( F " A ) ) )
27	3, 6, 26	syl2anc 693	. . 3 |- ( ph -> ( ( G " A ) C_ ( F " A ) <-> A. x e. A ( G ` x ) e. ( F " A ) ) )
28	25, 27	mpbird 247	. 2 |- ( ph -> ( G " A ) C_ ( F " A ) )
29	17, 28	eqssd 3620	1 |- ( ph -> ( F " A ) = ( G " A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   C_ wss 3574   dom cdm 5114   "cima 5117   Fun wfun 5882   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: (None)