Theorem fcnvgreu 26135

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 4952	. . . 4 |- ran A = dom `' A
2	1	eleq2i 2529	. . 3 |- ( Y e. ran A <-> Y e. dom `' A )
3		fgreu 26134	. . . 4 |- ( ( Fun `' A /\ Y e. dom `' A ) -> E! q e. `' A Y = ( 1st ` q ) )
4	3	adantll 713	. . 3 |- ( ( ( Rel A /\ Fun `' A ) /\ Y e. dom `' A ) -> E! q e. `' A Y = ( 1st ` q ) )
5	2, 4	sylan2b 475	. 2 |- ( ( ( Rel A /\ Fun `' A ) /\ Y e. ran A ) -> E! q e. `' A Y = ( 1st ` q ) )
6		cnvcnvss 5393	. . . . . 6 |- `' `' A C_ A
7		cnvssrndm 5460	. . . . . . . . . . 11 |- `' A C_ ( ran A X. dom A )
8	7	sseli 3453	. . . . . . . . . 10 |- ( q e. `' A -> q e. ( ran A X. dom A ) )
9		dfdm4 5133	. . . . . . . . . . 11 |- dom A = ran `' A
10	1, 9	xpeq12i 4963	. . . . . . . . . 10 |- ( ran A X. dom A ) = ( dom `' A X. ran `' A )
11	8, 10	syl6eleq 2549	. . . . . . . . 9 |- ( q e. `' A -> q e. ( dom `' A X. ran `' A ) )
12		2nd1st 6722	. . . . . . . . 9 |- ( q e. ( dom `' A X. ran `' A ) -> U. `' { q } = <. ( 2nd ` q ) , ( 1st ` q ) >. )
13	11, 12	syl 16	. . . . . . . 8 |- ( q e. `' A -> U. `' { q } = <. ( 2nd ` q ) , ( 1st ` q ) >. )
14	13	eqcomd 2459	. . . . . . 7 |- ( q e. `' A -> <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } )
15		relcnv 5307	. . . . . . . 8 |- Rel `' A
16		cnvf1olem 6773	. . . . . . . . 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 459	. . . . . . . 8 |- ( ( Rel `' A /\ ( q e. `' A /\ <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } ) ) -> <. ( 2nd ` q ) , ( 1st ` q ) >. e. `' `' A )
18	15, 17	mpan 670	. . . . . . 7 |- ( ( q e. `' A /\ <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } ) -> <. ( 2nd ` q ) , ( 1st ` q ) >. e. `' `' A )
19	14, 18	mpdan 668	. . . . . 6 |- ( q e. `' A -> <. ( 2nd ` q ) , ( 1st ` q ) >. e. `' `' A )
20	6, 19	sseldi 3455	. . . . 5 |- ( q e. `' A -> <. ( 2nd ` q ) , ( 1st ` q ) >. e. A )
21	20	adantl 466	. . . 4 |- ( ( ( Rel A /\ Fun `' A ) /\ q e. `' A ) -> <. ( 2nd ` q ) , ( 1st ` q ) >. e. A )
22		simpl 457	. . . . . . . 8 |- ( ( Rel A /\ Fun `' A ) -> Rel A )
23	22	adantr 465	. . . . . . 7 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> Rel A )
24		simpr 461	. . . . . . 7 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> p e. A )
25		relssdmrn 5459	. . . . . . . . . . 11 |- ( Rel A -> A C_ ( dom A X. ran A ) )
26	22, 25	syl 16	. . . . . . . . . 10 |- ( ( Rel A /\ Fun `' A ) -> A C_ ( dom A X. ran A ) )
27	26	sselda 3457	. . . . . . . . 9 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> p e. ( dom A X. ran A ) )
28		2nd1st 6722	. . . . . . . . 9 |- ( p e. ( dom A X. ran A ) -> U. `' { p } = <. ( 2nd ` p ) , ( 1st ` p ) >. )
29	27, 28	syl 16	. . . . . . . 8 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> U. `' { p } = <. ( 2nd ` p ) , ( 1st ` p ) >. )
30	29	eqcomd 2459	. . . . . . 7 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> <. ( 2nd ` p ) , ( 1st ` p ) >. = U. `' { p } )
31		cnvf1olem 6773	. . . . . . . 8 |- ( ( Rel A /\ ( p e. A /\ <. ( 2nd ` p ) , ( 1st ` p ) >. = U. `' { p } ) ) -> ( <. ( 2nd ` p ) , ( 1st ` p ) >. e. `' A /\ p = U. `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } ) )
32	31	simpld 459	. . . . . . 7 |- ( ( Rel A /\ ( p e. A /\ <. ( 2nd ` p ) , ( 1st ` p ) >. = U. `' { p } ) ) -> <. ( 2nd ` p ) , ( 1st ` p ) >. e. `' A )
33	23, 24, 30, 32	syl12anc 1217	. . . . . 6 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> <. ( 2nd ` p ) , ( 1st ` p ) >. e. `' A )
34	15	a1i 11	. . . . . . . . . 10 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> Rel `' A )
35		simplr 754	. . . . . . . . . 10 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> q e. `' A )
36	14	ad2antlr 726	. . . . . . . . . 10 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } )
37	16	simprd 463	. . . . . . . . . 10 |- ( ( Rel `' A /\ ( q e. `' A /\ <. ( 2nd ` q ) , ( 1st ` q ) >. = U. `' { q } ) ) -> q = U. `' { <. ( 2nd ` q ) , ( 1st ` q ) >. } )
38	34, 35, 36, 37	syl12anc 1217	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> q = U. `' { <. ( 2nd ` q ) , ( 1st ` q ) >. } )
39		simpr 461	. . . . . . . . . . . 12 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> p = <. ( 2nd ` q ) , ( 1st ` q ) >. )
40	39	sneqd 3990	. . . . . . . . . . 11 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> { p } = { <. ( 2nd ` q ) , ( 1st ` q ) >. } )
41	40	cnveqd 5116	. . . . . . . . . 10 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> `' { p } = `' { <. ( 2nd ` q ) , ( 1st ` q ) >. } )
42	41	unieqd 4202	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> U. `' { p } = U. `' { <. ( 2nd ` q ) , ( 1st ` q ) >. } )
43	29	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> U. `' { p } = <. ( 2nd ` p ) , ( 1st ` p ) >. )
44	38, 42, 43	3eqtr2d 2498	. . . . . . . 8 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> q = <. ( 2nd ` p ) , ( 1st ` p ) >. )
45	31	simprd 463	. . . . . . . . . . 11 |- ( ( Rel A /\ ( p e. A /\ <. ( 2nd ` p ) , ( 1st ` p ) >. = U. `' { p } ) ) -> p = U. `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
46	23, 24, 30, 45	syl12anc 1217	. . . . . . . . . 10 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> p = U. `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
47	46	ad2antrr 725	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> p = U. `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
48		simpr 461	. . . . . . . . . . . 12 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> q = <. ( 2nd ` p ) , ( 1st ` p ) >. )
49	48	sneqd 3990	. . . . . . . . . . 11 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> { q } = { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
50	49	cnveqd 5116	. . . . . . . . . 10 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> `' { q } = `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
51	50	unieqd 4202	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> U. `' { q } = U. `' { <. ( 2nd ` p ) , ( 1st ` p ) >. } )
52	13	ad2antlr 726	. . . . . . . . 9 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> U. `' { q } = <. ( 2nd ` q ) , ( 1st ` q ) >. )
53	47, 51, 52	3eqtr2d 2498	. . . . . . . 8 |- ( ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) /\ q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) -> p = <. ( 2nd ` q ) , ( 1st ` q ) >. )
54	44, 53	impbida 828	. . . . . . 7 |- ( ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) /\ q e. `' A ) -> ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) )
55	54	ralrimiva 2825	. . . . . 6 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> A. q e. `' A ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) )
56		biidd 237	. . . . . . . . . . 11 |- ( r = <. ( 2nd ` p ) , ( 1st ` p ) >. -> ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) )
57		eqeq2 2466	. . . . . . . . . . 11 |- ( r = <. ( 2nd ` p ) , ( 1st ` p ) >. -> ( q = r <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) )
58	56, 57	bibi12d 321	. . . . . . . . . 10 |- ( r = <. ( 2nd ` p ) , ( 1st ` p ) >. -> ( ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = r ) <-> ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) ) )
59	58	ralrimivw 2826	. . . . . . . . 9 |- ( r = <. ( 2nd ` p ) , ( 1st ` p ) >. -> A. q e. `' A ( ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = r ) <-> ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) ) )
60	59	r19.21bi 2913	. . . . . . . 8 |- ( ( r = <. ( 2nd ` p ) , ( 1st ` p ) >. /\ q e. `' A ) -> ( ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = r ) <-> ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = <. ( 2nd ` p ) , ( 1st ` p ) >. ) ) )
61	60	ralbidva 2839	. . . . . . 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 ) >. ) ) )
62	61	rspcev 3172	. . . . . 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 ) )
63	33, 55, 62	syl2anc 661	. . . . 5 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> E. r e. `' A A. q e. `' A ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. <-> q = r ) )
64		reu6 3248	. . . . 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 ) )
65	63, 64	sylibr 212	. . . 4 |- ( ( ( Rel A /\ Fun `' A ) /\ p e. A ) -> E! q e. `' A p = <. ( 2nd ` q ) , ( 1st ` q ) >. )
66		fvex 5802	. . . . . . 7 |- ( 2nd ` q ) e. _V
67		fvex 5802	. . . . . . 7 |- ( 1st ` q ) e. _V
68	66, 67	op2ndd 6691	. . . . . 6 |- ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. -> ( 2nd ` p ) = ( 1st ` q ) )
69	68	eqeq2d 2465	. . . . 5 |- ( p = <. ( 2nd ` q ) , ( 1st ` q ) >. -> ( Y = ( 2nd ` p ) <-> Y = ( 1st ` q ) ) )
70	69	adantl 466	. . . 4 |- ( ( ( Rel A /\ Fun `' A ) /\ p = <. ( 2nd ` q ) , ( 1st ` q ) >. ) -> ( Y = ( 2nd ` p ) <-> Y = ( 1st ` q ) ) )
71	21, 65, 70	reuxfr4d 26019	. . 3 |- ( ( Rel A /\ Fun `' A ) -> ( E! p e. A Y = ( 2nd ` p ) <-> E! q e. `' A Y = ( 1st ` q ) ) )
72	71	adantr 465	. 2 |- ( ( ( Rel A /\ Fun `' A ) /\ Y e. ran A ) -> ( E! p e. A Y = ( 2nd ` p ) <-> E! q e. `' A Y = ( 1st ` q ) ) )
73	5, 72	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 369   = wceq 1370   e. wcel 1758   A.wral 2795   E.wrex 2796   E!wreu 2797   C_ wss 3429   {csn 3978   <.cop 3984   U.cuni 4192   X. cxp 4939   `'ccnv 4940   dom cdm 4941   ran crn 4942   Rel wrel 4946   Fun wfun 5513   ` cfv 5519   1stc1st 6678   2ndc2nd 6679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-fv 5527  df-1st 6680  df-2nd 6681

This theorem is referenced by:  gsummpt2co  26387