Theorem cnvf1olem 6433

Description: Lemma for cnvf1o 6434. (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 533	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = U. `' { B } )
2		1st2nd 6388	. . . . . . . 8 |- ( ( Rel A /\ B e. A ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	2	adantrr 479	. . . . . . 7 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
4	3	sneqd 3707	. . . . . 6 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> { B } = { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
5	4	cnveqd 4936	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> `' { B } = `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
6	5	unieqd 3930	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { B } = U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
7		1stexg 6374	. . . . . 6 |- ( B e. A -> ( 1st ` B ) e. _V )
8		2ndexg 6375	. . . . . 6 |- ( B e. A -> ( 2nd ` B ) e. _V )
9		opswapg 5254	. . . . . 6 |- ( ( ( 1st ` B ) e. _V /\ ( 2nd ` B ) e. _V ) -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
10	7, 8, 9	syl2anc 411	. . . . 5 |- ( B e. A -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
11	10	ad2antrl 490	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >. )
12	1, 6, 11	3eqtrd 2271	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = <. ( 2nd ` B ) , ( 1st ` B ) >. )
13		simprl 531	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B e. A )
14	3, 13	eqeltrrd 2312	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A )
15		opelcnvg 4940	. . . . . 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 411	. . . . 5 |- ( B e. A -> ( <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A <-> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A ) )
17	16	ad2antrl 490	. . . 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 167	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A )
19	12, 18	eqeltrd 2311	. 2 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C e. `' A )
20		opswapg 5254	. . . . . 6 |- ( ( ( 2nd ` B ) e. _V /\ ( 1st ` B ) e. _V ) -> U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } = <. ( 1st ` B ) , ( 2nd ` B ) >. )
21	8, 7, 20	syl2anc 411	. . . . 5 |- ( B e. A -> U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } = <. ( 1st ` B ) , ( 2nd ` B ) >. )
22	21	eqcomd 2240	. . . 4 |- ( B e. A -> <. ( 1st ` B ) , ( 2nd ` B ) >. = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
23	22	ad2antrl 490	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
24	12	sneqd 3707	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> { C } = { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
25	24	cnveqd 4936	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> `' { C } = `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
26	25	unieqd 3930	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { C } = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
27	23, 3, 26	3eqtr4d 2277	. 2 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B = U. `' { C } )
28	19, 27	jca 306	1 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   {csn 3694   <.cop 3697   U.cuni 3919   `'ccnv 4753   Rel wrel 4759   ` cfv 5357   1stc1st 6345   2ndc2nd 6346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-1st 6347  df-2nd 6348

This theorem is referenced by:  cnvf1o  6434