Theorem fsneqrn 39403

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

Hypotheses

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

Assertion

Ref	Expression
fsneqrn	|- ( ph -> ( F = G <-> ( F ` A ) e. ran G ) )

Proof of Theorem fsneqrn

Step	Hyp	Ref	Expression
1		fsneqrn.f	. . . . . . 7 |- ( ph -> F Fn B )
2		dffn3 6054	. . . . . . 7 |- ( F Fn B <-> F : B --> ran F )
3	1, 2	sylib 208	. . . . . 6 |- ( ph -> F : B --> ran F )
4		fsneqrn.a	. . . . . . . 8 |- ( ph -> A e. V )
5		snidg 4206	. . . . . . . 8 |- ( A e. V -> A e. { A } )
6	4, 5	syl 17	. . . . . . 7 |- ( ph -> A e. { A } )
7		fsneqrn.b	. . . . . . . . 9 |- B = { A }
8	7	a1i 11	. . . . . . . 8 |- ( ph -> B = { A } )
9	8	eqcomd 2628	. . . . . . 7 |- ( ph -> { A } = B )
10	6, 9	eleqtrd 2703	. . . . . 6 |- ( ph -> A e. B )
11	3, 10	ffvelrnd 6360	. . . . 5 |- ( ph -> ( F ` A ) e. ran F )
12	11	adantr 481	. . . 4 |- ( ( ph /\ F = G ) -> ( F ` A ) e. ran F )
13		simpr 477	. . . . 5 |- ( ( ph /\ F = G ) -> F = G )
14	13	rneqd 5353	. . . 4 |- ( ( ph /\ F = G ) -> ran F = ran G )
15	12, 14	eleqtrd 2703	. . 3 |- ( ( ph /\ F = G ) -> ( F ` A ) e. ran G )
16	15	ex 450	. 2 |- ( ph -> ( F = G -> ( F ` A ) e. ran G ) )
17		simpr 477	. . . . . 6 |- ( ( ph /\ ( F ` A ) e. ran G ) -> ( F ` A ) e. ran G )
18		fsneqrn.g	. . . . . . . . . 10 |- ( ph -> G Fn B )
19		dffn2 6047	. . . . . . . . . 10 |- ( G Fn B <-> G : B --> _V )
20	18, 19	sylib 208	. . . . . . . . 9 |- ( ph -> G : B --> _V )
21	8	feq2d 6031	. . . . . . . . 9 |- ( ph -> ( G : B --> _V <-> G : { A } --> _V ) )
22	20, 21	mpbid 222	. . . . . . . 8 |- ( ph -> G : { A } --> _V )
23	4, 22	rnsnf 39370	. . . . . . 7 |- ( ph -> ran G = { ( G ` A ) } )
24	23	adantr 481	. . . . . 6 |- ( ( ph /\ ( F ` A ) e. ran G ) -> ran G = { ( G ` A ) } )
25	17, 24	eleqtrd 2703	. . . . 5 |- ( ( ph /\ ( F ` A ) e. ran G ) -> ( F ` A ) e. { ( G ` A ) } )
26		elsni 4194	. . . . 5 |- ( ( F ` A ) e. { ( G ` A ) } -> ( F ` A ) = ( G ` A ) )
27	25, 26	syl 17	. . . 4 |- ( ( ph /\ ( F ` A ) e. ran G ) -> ( F ` A ) = ( G ` A ) )
28	4	adantr 481	. . . . 5 |- ( ( ph /\ ( F ` A ) e. ran G ) -> A e. V )
29	1	adantr 481	. . . . 5 |- ( ( ph /\ ( F ` A ) e. ran G ) -> F Fn B )
30	18	adantr 481	. . . . 5 |- ( ( ph /\ ( F ` A ) e. ran G ) -> G Fn B )
31	28, 7, 29, 30	fsneq 39398	. . . 4 |- ( ( ph /\ ( F ` A ) e. ran G ) -> ( F = G <-> ( F ` A ) = ( G ` A ) ) )
32	27, 31	mpbird 247	. . 3 |- ( ( ph /\ ( F ` A ) e. ran G ) -> F = G )
33	32	ex 450	. 2 |- ( ph -> ( ( F ` A ) e. ran G -> F = G ) )
34	16, 33	impbid 202	1 |- ( ph -> ( F = G <-> ( F ` A ) e. ran G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   ran crn 5115   Fn wfn 5883   -->wf 5884   ` 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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  ssmapsn  39408