Theorem fsneq 39398

Description: Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
fsneq.a	|- ( ph -> A e. V )
fsneq.b	|- B = { A }
fsneq.f	|- ( ph -> F Fn B )
fsneq.g	|- ( ph -> G Fn B )

Assertion

Ref	Expression
fsneq	|- ( ph -> ( F = G <-> ( F ` A ) = ( G ` A ) ) )

Proof of Theorem fsneq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsneq.f	. . 3 |- ( ph -> F Fn B )
2		fsneq.g	. . 3 |- ( ph -> G Fn B )
3		eqfnfv 6311	. . 3 |- ( ( F Fn B /\ G Fn B ) -> ( F = G <-> A. x e. B ( F ` x ) = ( G ` x ) ) )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> ( F = G <-> A. x e. B ( F ` x ) = ( G ` x ) ) )
5		fsneq.a	. . . . . . . 8 |- ( ph -> A e. V )
6		snidg 4206	. . . . . . . 8 |- ( A e. V -> A e. { A } )
7	5, 6	syl 17	. . . . . . 7 |- ( ph -> A e. { A } )
8		fsneq.b	. . . . . . . . 9 |- B = { A }
9	8	eqcomi 2631	. . . . . . . 8 |- { A } = B
10	9	a1i 11	. . . . . . 7 |- ( ph -> { A } = B )
11	7, 10	eleqtrd 2703	. . . . . 6 |- ( ph -> A e. B )
12	11	adantr 481	. . . . 5 |- ( ( ph /\ A. x e. B ( F ` x ) = ( G ` x ) ) -> A e. B )
13		simpr 477	. . . . 5 |- ( ( ph /\ A. x e. B ( F ` x ) = ( G ` x ) ) -> A. x e. B ( F ` x ) = ( G ` x ) )
14		fveq2 6191	. . . . . . 7 |- ( x = A -> ( F ` x ) = ( F ` A ) )
15		fveq2 6191	. . . . . . 7 |- ( x = A -> ( G ` x ) = ( G ` A ) )
16	14, 15	eqeq12d 2637	. . . . . 6 |- ( x = A -> ( ( F ` x ) = ( G ` x ) <-> ( F ` A ) = ( G ` A ) ) )
17	16	rspcva 3307	. . . . 5 |- ( ( A e. B /\ A. x e. B ( F ` x ) = ( G ` x ) ) -> ( F ` A ) = ( G ` A ) )
18	12, 13, 17	syl2anc 693	. . . 4 |- ( ( ph /\ A. x e. B ( F ` x ) = ( G ` x ) ) -> ( F ` A ) = ( G ` A ) )
19	18	ex 450	. . 3 |- ( ph -> ( A. x e. B ( F ` x ) = ( G ` x ) -> ( F ` A ) = ( G ` A ) ) )
20		simpl 473	. . . . . . 7 |- ( ( ( F ` A ) = ( G ` A ) /\ x e. B ) -> ( F ` A ) = ( G ` A ) )
21	8	eleq2i 2693	. . . . . . . . . . 11 |- ( x e. B <-> x e. { A } )
22	21	biimpi 206	. . . . . . . . . 10 |- ( x e. B -> x e. { A } )
23		velsn 4193	. . . . . . . . . 10 |- ( x e. { A } <-> x = A )
24	22, 23	sylib 208	. . . . . . . . 9 |- ( x e. B -> x = A )
25	24	fveq2d 6195	. . . . . . . 8 |- ( x e. B -> ( F ` x ) = ( F ` A ) )
26	25	adantl 482	. . . . . . 7 |- ( ( ( F ` A ) = ( G ` A ) /\ x e. B ) -> ( F ` x ) = ( F ` A ) )
27	24	fveq2d 6195	. . . . . . . 8 |- ( x e. B -> ( G ` x ) = ( G ` A ) )
28	27	adantl 482	. . . . . . 7 |- ( ( ( F ` A ) = ( G ` A ) /\ x e. B ) -> ( G ` x ) = ( G ` A ) )
29	20, 26, 28	3eqtr4d 2666	. . . . . 6 |- ( ( ( F ` A ) = ( G ` A ) /\ x e. B ) -> ( F ` x ) = ( G ` x ) )
30	29	adantll 750	. . . . 5 |- ( ( ( ph /\ ( F ` A ) = ( G ` A ) ) /\ x e. B ) -> ( F ` x ) = ( G ` x ) )
31	30	ralrimiva 2966	. . . 4 |- ( ( ph /\ ( F ` A ) = ( G ` A ) ) -> A. x e. B ( F ` x ) = ( G ` x ) )
32	31	ex 450	. . 3 |- ( ph -> ( ( F ` A ) = ( G ` A ) -> A. x e. B ( F ` x ) = ( G ` x ) ) )
33	19, 32	impbid 202	. 2 |- ( ph -> ( A. x e. B ( F ` x ) = ( G ` x ) <-> ( F ` A ) = ( G ` A ) ) )
34	4, 33	bitrd 268	1 |- ( ph -> ( F = G <-> ( F ` A ) = ( G ` A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {csn 4177   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-8 1992  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-pow 4843  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-csb 3534  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-mpt 4730  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:  fsneqrn  39403  unirnmapsn  39406