Theorem fsnex 6538

Description: Relate a function with a singleton as domain and one variable. (Contributed by Thierry Arnoux, 12-Jul-2020.)

Hypothesis

Ref	Expression
fsnex.1	|- ( x = ( f ` A ) -> ( ps <-> ph ) )

Assertion

Ref	Expression
fsnex	|- ( A e. V -> ( E. f ( f : { A } --> D /\ ph ) <-> E. x e. D ps ) )

Distinct variable groups:   A, f, x   D, f, x   f, V, x   ps, f   ph, x

Allowed substitution hints:   ph( f)   ps( x)

Proof of Theorem fsnex

Step	Hyp	Ref	Expression
1		fsn2g 6405	. . . . . . . 8 |- ( A e. V -> ( f : { A } --> D <-> ( ( f ` A ) e. D /\ f = { <. A , ( f ` A ) >. } ) ) )
2	1	simprbda 653	. . . . . . 7 |- ( ( A e. V /\ f : { A } --> D ) -> ( f ` A ) e. D )
3	2	adantrr 753	. . . . . 6 |- ( ( A e. V /\ ( f : { A } --> D /\ ph ) ) -> ( f ` A ) e. D )
4		fsnex.1	. . . . . . 7 |- ( x = ( f ` A ) -> ( ps <-> ph ) )
5	4	adantl 482	. . . . . 6 |- ( ( ( A e. V /\ ( f : { A } --> D /\ ph ) ) /\ x = ( f ` A ) ) -> ( ps <-> ph ) )
6		simprr 796	. . . . . 6 |- ( ( A e. V /\ ( f : { A } --> D /\ ph ) ) -> ph )
7	3, 5, 6	rspcedvd 3317	. . . . 5 |- ( ( A e. V /\ ( f : { A } --> D /\ ph ) ) -> E. x e. D ps )
8	7	ex 450	. . . 4 |- ( A e. V -> ( ( f : { A } --> D /\ ph ) -> E. x e. D ps ) )
9	8	exlimdv 1861	. . 3 |- ( A e. V -> ( E. f ( f : { A } --> D /\ ph ) -> E. x e. D ps ) )
10	9	imp 445	. 2 |- ( ( A e. V /\ E. f ( f : { A } --> D /\ ph ) ) -> E. x e. D ps )
11		nfv 1843	. . . 4 |- F/ x A e. V
12		nfre1 3005	. . . 4 |- F/ x E. x e. D ps
13	11, 12	nfan 1828	. . 3 |- F/ x ( A e. V /\ E. x e. D ps )
14		vex 3203	. . . . . . . . 9 |- x e. _V
15		f1osng 6177	. . . . . . . . 9 |- ( ( A e. V /\ x e. _V ) -> { <. A , x >. } : { A } -1-1-onto-> { x } )
16	14, 15	mpan2 707	. . . . . . . 8 |- ( A e. V -> { <. A , x >. } : { A } -1-1-onto-> { x } )
17	16	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) -> { <. A , x >. } : { A } -1-1-onto-> { x } )
18		f1of 6137	. . . . . . 7 |- ( { <. A , x >. } : { A } -1-1-onto-> { x } -> { <. A , x >. } : { A } --> { x } )
19	17, 18	syl 17	. . . . . 6 |- ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) -> { <. A , x >. } : { A } --> { x } )
20		simplr 792	. . . . . . 7 |- ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) -> x e. D )
21	20	snssd 4340	. . . . . 6 |- ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) -> { x } C_ D )
22		fss 6056	. . . . . 6 |- ( ( { <. A , x >. } : { A } --> { x } /\ { x } C_ D ) -> { <. A , x >. } : { A } --> D )
23	19, 21, 22	syl2anc 693	. . . . 5 |- ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) -> { <. A , x >. } : { A } --> D )
24		fvsng 6447	. . . . . . . 8 |- ( ( A e. V /\ x e. _V ) -> ( { <. A , x >. } ` A ) = x )
25	14, 24	mpan2 707	. . . . . . 7 |- ( A e. V -> ( { <. A , x >. } ` A ) = x )
26	25	eqcomd 2628	. . . . . 6 |- ( A e. V -> x = ( { <. A , x >. } ` A ) )
27	26	ad3antrrr 766	. . . . 5 |- ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) -> x = ( { <. A , x >. } ` A ) )
28		snex 4908	. . . . . 6 |- { <. A , x >. } e. _V
29		feq1 6026	. . . . . . 7 |- ( f = { <. A , x >. } -> ( f : { A } --> D <-> { <. A , x >. } : { A } --> D ) )
30		fveq1 6190	. . . . . . . 8 |- ( f = { <. A , x >. } -> ( f ` A ) = ( { <. A , x >. } ` A ) )
31	30	eqeq2d 2632	. . . . . . 7 |- ( f = { <. A , x >. } -> ( x = ( f ` A ) <-> x = ( { <. A , x >. } ` A ) ) )
32	29, 31	anbi12d 747	. . . . . 6 |- ( f = { <. A , x >. } -> ( ( f : { A } --> D /\ x = ( f ` A ) ) <-> ( { <. A , x >. } : { A } --> D /\ x = ( { <. A , x >. } ` A ) ) ) )
33	28, 32	spcev 3300	. . . . 5 |- ( ( { <. A , x >. } : { A } --> D /\ x = ( { <. A , x >. } ` A ) ) -> E. f ( f : { A } --> D /\ x = ( f ` A ) ) )
34	23, 27, 33	syl2anc 693	. . . 4 |- ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) -> E. f ( f : { A } --> D /\ x = ( f ` A ) ) )
35		simprl 794	. . . . . . 7 |- ( ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) /\ ( f : { A } --> D /\ x = ( f ` A ) ) ) -> f : { A } --> D )
36		simpllr 799	. . . . . . . . 9 |- ( ( ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) /\ ( f : { A } --> D /\ x = ( f ` A ) ) ) /\ f : { A } --> D ) -> ps )
37		simplrr 801	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) /\ ( f : { A } --> D /\ x = ( f ` A ) ) ) /\ f : { A } --> D ) -> x = ( f ` A ) )
38	37, 4	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) /\ ( f : { A } --> D /\ x = ( f ` A ) ) ) /\ f : { A } --> D ) -> ( ps <-> ph ) )
39	36, 38	mpbid 222	. . . . . . . 8 |- ( ( ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) /\ ( f : { A } --> D /\ x = ( f ` A ) ) ) /\ f : { A } --> D ) -> ph )
40	35, 39	mpdan 702	. . . . . . 7 |- ( ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) /\ ( f : { A } --> D /\ x = ( f ` A ) ) ) -> ph )
41	35, 40	jca 554	. . . . . 6 |- ( ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) /\ ( f : { A } --> D /\ x = ( f ` A ) ) ) -> ( f : { A } --> D /\ ph ) )
42	41	ex 450	. . . . 5 |- ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) -> ( ( f : { A } --> D /\ x = ( f ` A ) ) -> ( f : { A } --> D /\ ph ) ) )
43	42	eximdv 1846	. . . 4 |- ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) -> ( E. f ( f : { A } --> D /\ x = ( f ` A ) ) -> E. f ( f : { A } --> D /\ ph ) ) )
44	34, 43	mpd 15	. . 3 |- ( ( ( ( A e. V /\ E. x e. D ps ) /\ x e. D ) /\ ps ) -> E. f ( f : { A } --> D /\ ph ) )
45		simpr 477	. . 3 |- ( ( A e. V /\ E. x e. D ps ) -> E. x e. D ps )
46	13, 44, 45	r19.29af 3076	. 2 |- ( ( A e. V /\ E. x e. D ps ) -> E. f ( f : { A } --> D /\ ph ) )
47	10, 46	impbida 877	1 |- ( A e. V -> ( E. f ( f : { A } --> D /\ ph ) <-> E. x e. D ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   -->wf 5884   -1-1-onto->wf1o 5887   ` 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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by: (None)