Theorem cnvf1olem 7275

Description: Lemma for cnvf1o 7276. (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 796	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = U. `' { B } )
2		1st2nd 7214	. . . . . . . . 9 |- ( ( Rel A /\ B e. A ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	2	adantrr 753	. . . . . . . 8 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
4	3	sneqd 4189	. . . . . . 7 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> { B } = { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
5	4	cnveqd 5298	. . . . . 6 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> `' { B } = `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
6	5	unieqd 4446	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { B } = U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
7	1, 6	eqtrd 2656	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } )
8		opswap 5622	. . . 4 |- U. `' { <. ( 1st ` B ) , ( 2nd ` B ) >. } = <. ( 2nd ` B ) , ( 1st ` B ) >.
9	7, 8	syl6eq 2672	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C = <. ( 2nd ` B ) , ( 1st ` B ) >. )
10		simprl 794	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B e. A )
11	3, 10	eqeltrrd 2702	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A )
12		fvex 6201	. . . . 5 |- ( 2nd ` B ) e. _V
13		fvex 6201	. . . . 5 |- ( 1st ` B ) e. _V
14	12, 13	opelcnv 5304	. . . 4 |- ( <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A <-> <. ( 1st ` B ) , ( 2nd ` B ) >. e. A )
15	11, 14	sylibr 224	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> <. ( 2nd ` B ) , ( 1st ` B ) >. e. `' A )
16	9, 15	eqeltrd 2701	. 2 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> C e. `' A )
17		opswap 5622	. . . 4 |- U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } = <. ( 1st ` B ) , ( 2nd ` B ) >.
18	17	eqcomi 2631	. . 3 |- <. ( 1st ` B ) , ( 2nd ` B ) >. = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. }
19	9	sneqd 4189	. . . . 5 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> { C } = { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
20	19	cnveqd 5298	. . . 4 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> `' { C } = `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
21	20	unieqd 4446	. . 3 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> U. `' { C } = U. `' { <. ( 2nd ` B ) , ( 1st ` B ) >. } )
22	18, 3, 21	3eqtr4a 2682	. 2 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> B = U. `' { C } )
23	16, 22	jca 554	1 |- ( ( Rel A /\ ( B e. A /\ C = U. `' { B } ) ) -> ( C e. `' A /\ B = U. `' { C } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   U.cuni 4436   `'ccnv 5113   Rel wrel 5119   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  cnvf1o  7276  fcnvgreu  29472