Theorem fcnvgreu 27957

Description: If the converse of a relation A is a function, exactly one point of its graph has a given second element (that is, function value) (Contributed by Thierry Arnoux, 1-Apr-2018.)

Assertion

Ref	Expression
fcnvgreu	|- ( ( ( Rel A /\ Fun `' A ) /\ Y e. ran A ) -> E! p e. A Y = ( 2nd ` p ) )

Distinct variable groups:   A, p   Y, p

Proof of Theorem fcnvgreu

Dummy variables q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rn 4834	. . . 4 |- ran A = dom `' A
2	1	eleq2i 2480	. . 3 |- ( Y e. ran A <-> Y e. dom `' A )
3		fgreu 27956	. . . 4 |- ( ( Fun `' A /\ Y e. dom `' A ) -> E! q e. `' A Y = ( 1st ` q ) )
4	3	adantll 712	. . 3 |- ( ( ( Rel A /\ Fun `' A ) /\ Y e. dom `' A ) -> E! q e. `' A Y = ( 1st ` q ) )
5	2, 4	sylan2b 473	. 2 |- ( ( ( Rel A /\ Fun `' A ) /\ Y e. ran A ) -> E! q e. `' A Y = ( 1st ` q ) )
6		cnvcnvss 5278	. . . . . 6 |- `' `' A C_ A
7		cnvssrndm 5345	. . . . . . . . . . 11 |- `' A C_ ( ran A X. dom A )
8	7	sseli 3438	. . . . . . . . . 10 |- ( q e. `' A -> q e. ( ran A X. dom A ) )
9		dfdm4 5016	. . . . . . . . . . 11 |- dom A = ran `' A
10	1, 9	xpeq12i 4845	. . . . . . . . . 10 |- ( ran A X. dom A ) = ( dom `' A X. ran `' A )
11	8, 10	syl6eleq 2500	. . . . . . . . 9 |- ( q e. `' A -> q e. ( dom `' A X. ran `' A ) )
12		2nd1st 6829	. . . . . . . . 9 |- ( q e. ( dom `' A X. ran `' A ) -> U. `' { q } = <. ( 2nd ` q ) , ( 1st ` q ) >. )
13	11, 12	syl 17	. . . . . . . 8 |- ( q e. `' A -> U. `' { q } = <. ( 2nd ` q ) , ( 1st ` q ) >. )
14	13	eqcomd 2410	. . . . . . 7 |- ( q e. `' A -> <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } )
15		relcnv 5195	. . . . . . . 8 |- Rel `' A
16		cnvf1olem 6882	. . . . . . . . 9 |- ( ( Rel `' A /\ ( q e. `' A /\ <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } ) ) -> ( <. ( 2nd ` q ) , ( 1st ` q ) >. e. `' `' A /\ q = U. `' { <. ( 2nd ` q ) , ( 1st ` q ) >. } ) )
17	16	simpld 457	. . . . . . . 8 |- ( ( Rel `' A /\ ( q e. `' A /\ <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } ) ) -> <. ( 2nd ` q ) , ( 1st ` q ) >. e. `' `' A )
18	15, 17	mpan 668	. . . . . . 7 |- ( ( q e. `' A /\ <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } ) -> <. ( 2nd ` q ) , ( 1st ` q ) >. e. `' `' A )
19	14, 18	mpdan 666	. . . . . 6 |- ( q e. `' A -> <. ( 2nd ` q ) , ( 1st ` q ) >. e. `' `' A )
20	6, 19	sseldi 3440	. . . . 5 |- ( q e. `' A -> <. ( 2nd ` q ) , ( 1st ` q ) >. e. A )
21	20	adantl 464	. . . 4 |- ( ( ( Rel A /\ Fun `' A ) /\ q e. `' A ) -> <. ( 2nd ` q ) , ( 1st ` q ) >. e. A )
22		simpll 752	. . . . . . 7 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> Rel A )
23		simpr 459	. . . . . . 7 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> p e. A )
24		relssdmrn 5344	. . . . . . . . . . 11 |- ( Rel A -> A C_ ( dom A X. ran A ) )
25	24	adantr 463	. . . . . . . . . 10 |- ( ( Rel A /\ Fun `' A ) -> A C_ ( dom A X. ran A ) )
26	25	sselda 3442	. . . . . . . . 9 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> p e. ( dom A X. ran A ) )
27		2nd1st 6829	. . . . . . . . 9 |- ( p e. ( dom A X. ran A ) -> U. `' { p } = <. ( 2nd ` p ) , ( 1st ` p ) >. )
28	26, 27	syl 17	. . . . . . . 8 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> U. `' { p } = <. ( 2nd ` p ) , ( 1st ` p ) >. )
29	28	eqcomd 2410	. . . . . . 7 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> <. ( 2nd ` p ) , ( 1st ` p ) >. = U. `' { p } )
30		cnvf1olem 6882	. . . . . . . 8 |- ( ( Rel A /\ ( p e. A /\ <. ( 2nd ` p ) , ( 1st ` p ) >. = U. `' { p } ) ) -> ( <. ( 2nd ` p ) , ( 1st ` p ) >. e. `' A /\ p = U. `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } ) )
31	30	simpld 457	. . . . . . 7 |- ( ( Rel A /\ ( p e. A /\ <. ( 2nd ` p ) , ( 1st ` p ) >. = U. `' { p } ) ) -> <. ( 2nd ` p ) , ( 1st ` p ) >. e. `' A )
32	22, 23, 29, 31	syl12anc 1228	. . . . . 6 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> <. ( 2nd ` p ) , ( 1st ` p ) >. e. `' A )
33	15	a1i 11	. . . . . . . . . 10 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> Rel `' A )
34		simplr 754	. . . . . . . . . 10 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> q e. `' A )
35	14	ad2antlr 725	. . . . . . . . . 10 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } )
36	16	simprd 461	. . . . . . . . . 10 |- ( ( Rel `' A /\ ( q e. `' A /\ <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } ) ) -> q = U. `' { <. ( 2nd ` q ) , ( 1st ` q ) >. } )
37	33, 34, 35, 36	syl12anc 1228	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> q = U. `' { <. ( 2nd ` q ) , ( 1st ` q ) >. } )
38		simpr 459	. . . . . . . . . . . 12 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> p = <. ( 2nd ` q ) , ( 1st ` q ) >. )
39	38	sneqd 3984	. . . . . . . . . . 11 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> { p } = { <. ( 2nd ` q ) , ( 1st ` q ) >. } )
40	39	cnveqd 4999	. . . . . . . . . 10 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> `' { p } = `' { <. ( 2nd ` q ) , ( 1st ` q ) >. } )
41	40	unieqd 4201	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> U. `' { p } = U. `' { <. ( 2nd ` q ) , ( 1st ` q ) >. } )
42	28	ad2antrr 724	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> U. `' { p } = <. ( 2nd ` p ) , ( 1st ` p ) >. )
43	37, 41, 42	3eqtr2d 2449	. . . . . . . 8 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> q = <. ( 2nd ` p ) , ( 1st ` p ) >. )
44	30	simprd 461	. . . . . . . . . . 11 |- ( ( Rel A /\ ( p e. A /\ <. ( 2nd ` p ) , ( 1st ` p ) >. = U. `' { p } ) ) -> p = U. `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
45	22, 23, 29, 44	syl12anc 1228	. . . . . . . . . 10 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> p = U. `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
46	45	ad2antrr 724	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> p = U. `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
47		simpr 459	. . . . . . . . . . . 12 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> q = <. ( 2nd ` p ) , ( 1st ` p ) >. )
48	47	sneqd 3984	. . . . . . . . . . 11 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> { q } = { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
49	48	cnveqd 4999	. . . . . . . . . 10 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> `' { q } = `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
50	49	unieqd 4201	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> U. `' { q } = U. `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
51	13	ad2antlr 725	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> U. `' { q } = <. ( 2nd ` q ) , ( 1st ` q ) >. )
52	46, 50, 51	3eqtr2d 2449	. . . . . . . 8 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> p = <. ( 2nd ` q ) , ( 1st ` q ) >. )
53	43, 52	impbida 833	. . . . . . 7 |- ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) -> ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) )
54	53	ralrimiva 2818	. . . . . 6 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> A. q e. `' A ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) )
55		eqeq2 2417	. . . . . . . . 9 |- ( r = <. ( 2nd ` p ) , ( 1st ` p ) >. -> ( q = r <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) )
56	55	bibi2d 316	. . . . . . . 8 |- ( r = <. ( 2nd ` p ) , ( 1st ` p ) >. -> ( ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = r ) <-> ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) ) )
57	56	ralbidv 2843	. . . . . . 7 |- ( r = <. ( 2nd ` p ) , ( 1st ` p ) >. -> ( A. q e. `' A ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = r ) <-> A. q e. `' A ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) ) )
58	57	rspcev 3160	. . . . . 6 |- ( ( <. ( 2nd ` p ) , ( 1st ` p ) >. e. `' A /\ A. q e. `' A ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) ) -> E. r e. `' A A. q e. `' A ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = r ) )
59	32, 54, 58	syl2anc 659	. . . . 5 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> E. r e. `' A A. q e. `' A ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = r ) )
60		reu6 3238	. . . . 5 |- ( E! q e. `' A p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> E. r e. `' A A. q e. `' A ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = r ) )
61	59, 60	sylibr 212	. . . 4 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> E! q e. `' A p = <. ( 2nd ` q ) , ( 1st ` q ) >. )
62		fvex 5859	. . . . . . 7 |- ( 2nd ` q ) e. _V
63		fvex 5859	. . . . . . 7 |- ( 1st ` q ) e. _V
64	62, 63	op2ndd 6795	. . . . . 6 |- ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. -> ( 2nd ` p ) = ( 1st ` q ) )
65	64	eqeq2d 2416	. . . . 5 |- ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. -> ( Y = ( 2nd ` p ) <-> Y = ( 1st ` q ) ) )
66	65	adantl 464	. . . 4 |- ( ( ( Rel A /\ Fun `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> ( Y = ( 2nd ` p ) <-> Y = ( 1st ` q ) ) )
67	21, 61, 66	reuxfr4d 27804	. . 3 |- ( ( Rel A /\ Fun `' A ) -> ( E! p e. A Y = ( 2nd ` p ) <-> E! q e. `' A Y = ( 1st ` q ) ) )
68	67	adantr 463	. 2 |- ( ( ( Rel A /\ Fun `' A ) /\ Y e. ran A ) -> ( E! p e. A Y = ( 2nd ` p ) <-> E! q e. `' A Y = ( 1st ` q ) ) )
69	5, 68	mpbird 232	1 |- ( ( ( Rel A /\ Fun `' A ) /\ Y e. ran A ) -> E! p e. A Y = ( 2nd ` p ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1405   e. wcel 1842   A.wral 2754   E.wrex 2755   E!wreu 2756   C_ wss 3414   {csn 3972   <.cop 3978   U.cuni 4191   X. cxp 4821   `'ccnv 4822   dom cdm 4823   ran crn 4824   Rel wrel 4828   Fun wfun 5563   ` cfv 5569   1stc1st 6782   2ndc2nd 6783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-iota 5533  df-fun 5571  df-fn 5572  df-fv 5577  df-1st 6784  df-2nd 6785

This theorem is referenced by:  gsummpt2co  28222