Theorem cnvf1olem 6073

Description: Lemma for cnvf1o 6074. (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 504	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = U. `' { B } )
2		1st2nd 6031	. . . . . . . 8 |- ( ( Rel A /\ B e. A ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	2	adantrr 468	. . . . . . 7 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
4	3	sneqd 3504	. . . . . 6 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> { B } = { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
5	4	cnveqd 4673	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> `' { B } = `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
6	5	unieqd 3711	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { B } = U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
7		1stexg 6017	. . . . . 6 |- ( B e. A -> ( 1st ` B ) e. _V )
8		2ndexg 6018	. . . . . 6 |- ( B e. A -> ( 2nd ` B ) e. _V )
9		opswapg 4981	. . . . . 6 |- ( ( ( 1st ` B ) e. _V /\ ( 2nd ` B ) e. _V ) -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
10	7, 8, 9	syl2anc 406	. . . . 5 |- ( B e. A -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
11	10	ad2antrl 479	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
12	1, 6, 11	3eqtrd 2149	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = <. ( 2nd ` B ) , ( 1st ` B ) >. )
13		simprl 503	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B e. A )
14	3, 13	eqeltrrd 2190	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A )
15		opelcnvg 4677	. . . . . 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 406	. . . . 5 |- ( B e. A -> ( <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A <-> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A ) )
17	16	ad2antrl 479	. . . 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 166	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A )
19	12, 18	eqeltrd 2189	. 2 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C e. `' A )
20		opswapg 4981	. . . . . 6 |- ( ( ( 2nd ` B ) e. _V /\ ( 1st ` B ) e. _V ) -> U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } = <. ( 1st ` B ) , ( 2nd ` B ) >. )
21	8, 7, 20	syl2anc 406	. . . . 5 |- ( B e. A -> U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } = <. ( 1st ` B ) , ( 2nd ` B ) >. )
22	21	eqcomd 2118	. . . 4 |- ( B e. A -> <. ( 1st ` B ) , ( 2nd ` B ) >. = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
23	22	ad2antrl 479	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
24	12	sneqd 3504	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> { C } = { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
25	24	cnveqd 4673	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> `' { C } = `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
26	25	unieqd 3711	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { C } = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
27	23, 3, 26	3eqtr4d 2155	. 2 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B = U. `' { C } )
28	19, 27	jca 302	1 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1312   e. wcel 1461   _Vcvv 2655   {csn 3491   <.cop 3494   U.cuni 3700   `'ccnv 4496   Rel wrel 4502   ` cfv 5079   1stc1st 5988   2ndc2nd 5989

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-fo 5085  df-fv 5087  df-1st 5990  df-2nd 5991

This theorem is referenced by:  cnvf1o  6074