Theorem cnvf1olem 5865

Description: Lemma for cnvf1o 5866. (Contributed by Mario Carneiro, 27-Apr-2014.)

Assertion

Ref	Expression
cnvf1olem	|- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )

Proof of Theorem cnvf1olem

Step	Hyp	Ref	Expression
1		simprr 498	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = U. `' { B } )
2		1st2nd 5827	. . . . . . . 8 |- ( ( Rel A /\ B e. A ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	2	adantrr 462	. . . . . . 7 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
4	3	sneqd 3411	. . . . . 6 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> { B } = { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
5	4	cnveqd 4529	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> `' { B } = `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
6	5	unieqd 3612	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { B } = U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
7		1stexg 5814	. . . . . 6 |- ( B e. A -> ( 1st ` B ) e. _V )
8		2ndexg 5815	. . . . . 6 |- ( B e. A -> ( 2nd ` B ) e. _V )
9		opswapg 4827	. . . . . 6 |- ( ( ( 1st ` B ) e. _V /\ ( 2nd ` B ) e. _V ) -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
10	7, 8, 9	syl2anc 403	. . . . 5 |- ( B e. A -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
11	10	ad2antrl 473	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
12	1, 6, 11	3eqtrd 2117	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = <. ( 2nd ` B ) , ( 1st ` B ) >. )
13		simprl 497	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B e. A )
14	3, 13	eqeltrrd 2156	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A )
15		opelcnvg 4533	. . . . . 6 |- ( ( ( 2nd ` B ) e. _V /\ ( 1st ` B ) e. _V ) -> ( <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A <-> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A ) )
16	8, 7, 15	syl2anc 403	. . . . 5 |- ( B e. A -> ( <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A <-> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A ) )
17	16	ad2antrl 473	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A <-> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A ) )
18	14, 17	mpbird 165	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A )
19	12, 18	eqeltrd 2155	. 2 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C e. `' A )
20		opswapg 4827	. . . . . 6 |- ( ( ( 2nd ` B ) e. _V /\ ( 1st ` B ) e. _V ) -> U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } = <. ( 1st ` B ) , ( 2nd ` B ) >. )
21	8, 7, 20	syl2anc 403	. . . . 5 |- ( B e. A -> U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } = <. ( 1st ` B ) , ( 2nd ` B ) >. )
22	21	eqcomd 2086	. . . 4 |- ( B e. A -> <. ( 1st ` B ) , ( 2nd ` B ) >. = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
23	22	ad2antrl 473	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
24	12	sneqd 3411	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> { C } = { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
25	24	cnveqd 4529	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> `' { C } = `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
26	25	unieqd 3612	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { C } = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
27	23, 3, 26	3eqtr4d 2123	. 2 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B = U. `' { C } )
28	19, 27	jca 300	1 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   {csn 3398   <.cop 3401   U.cuni 3601   `'ccnv 4362   Rel wrel 4368   ` cfv 4922   1stc1st 5785   2ndc2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-1st 5787  df-2nd 5788

This theorem is referenced by:  cnvf1o  5866