Theorem cnvf1olem 6192

Description: Lemma for cnvf1o 6193. (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 522	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = U. `' { B } )
2		1st2nd 6149	. . . . . . . 8 |- ( ( Rel A /\ B e. A ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	2	adantrr 471	. . . . . . 7 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
4	3	sneqd 3589	. . . . . 6 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> { B } = { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
5	4	cnveqd 4780	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> `' { B } = `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
6	5	unieqd 3800	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { B } = U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
7		1stexg 6135	. . . . . 6 |- ( B e. A -> ( 1st ` B ) e. _V )
8		2ndexg 6136	. . . . . 6 |- ( B e. A -> ( 2nd ` B ) e. _V )
9		opswapg 5090	. . . . . 6 |- ( ( ( 1st ` B ) e. _V /\ ( 2nd ` B ) e. _V ) -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
10	7, 8, 9	syl2anc 409	. . . . 5 |- ( B e. A -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
11	10	ad2antrl 482	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
12	1, 6, 11	3eqtrd 2202	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = <. ( 2nd ` B ) , ( 1st ` B ) >. )
13		simprl 521	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B e. A )
14	3, 13	eqeltrrd 2244	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A )
15		opelcnvg 4784	. . . . . 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 409	. . . . 5 |- ( B e. A -> ( <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A <-> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A ) )
17	16	ad2antrl 482	. . . 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 2243	. 2 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C e. `' A )
20		opswapg 5090	. . . . . 6 |- ( ( ( 2nd ` B ) e. _V /\ ( 1st ` B ) e. _V ) -> U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } = <. ( 1st ` B ) , ( 2nd ` B ) >. )
21	8, 7, 20	syl2anc 409	. . . . 5 |- ( B e. A -> U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } = <. ( 1st ` B ) , ( 2nd ` B ) >. )
22	21	eqcomd 2171	. . . 4 |- ( B e. A -> <. ( 1st ` B ) , ( 2nd ` B ) >. = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
23	22	ad2antrl 482	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
24	12	sneqd 3589	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> { C } = { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
25	24	cnveqd 4780	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> `' { C } = `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
26	25	unieqd 3800	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { C } = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
27	23, 3, 26	3eqtr4d 2208	. 2 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B = U. `' { C } )
28	19, 27	jca 304	1 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   {csn 3576   <.cop 3579   U.cuni 3789   `'ccnv 4603   Rel wrel 4609   ` cfv 5188   1stc1st 6106   2ndc2nd 6107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-1st 6108  df-2nd 6109

This theorem is referenced by:  cnvf1o  6193