Theorem cnvf1olem 5787

Description: Lemma for cnvf1o 5788. (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 484	. . . 4 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> C = U. `' {B })
2		1st2nd 5749	. . . . . . . 8 |- (( Rel A /\ B e. A) -> B = <.( 1st ` B) , ( 2nd ` B) >.)
3	2	adantrr 448	. . . . . . 7 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> B = <.( 1st ` B) , ( 2nd ` B) >.)
4	3	sneqd 3380	. . . . . 6 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> {B } = { <.( 1st ` B) , ( 2nd ` B) >. })
5	4	cnveqd 4454	. . . . 5 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> `' {B } = `' { <.( 1st ` B) , ( 2nd ` B) >. })
6	5	unieqd 3582	. . . 4 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> U. `' {B } = U. `' { <.( 1st ` B) , ( 2nd ` B) >. })
7		1stexg 5736	. . . . . 6 |- (B e. A -> ( 1st ` B) e. _V)
8		2ndexg 5737	. . . . . 6 |- (B e. A -> ( 2nd ` B) e. _V)
9		opswapg 4750	. . . . . 6 |- ((( 1st ` B) e. _V /\ ( 2nd ` B) e. _V) -> U. `' { <.( 1st ` B) , ( 2nd ` B) >. } = <.( 2nd ` B) , ( 1st ` B) >.)
10	7, 8, 9	syl2anc 391	. . . . 5 |- (B e. A -> U. `' { <.( 1st ` B) , ( 2nd ` B) >. } = <.( 2nd ` B) , ( 1st ` B) >.)
11	10	ad2antrl 459	. . . 4 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> U. `' { <.( 1st ` B) , ( 2nd ` B) >. } = <.( 2nd ` B) , ( 1st ` B) >.)
12	1, 6, 11	3eqtrd 2073	. . 3 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> C = <.( 2nd ` B) , ( 1st ` B) >.)
13		simprl 483	. . . . 5 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> B e. A)
14	3, 13	eqeltrrd 2112	. . . 4 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> <.( 1st ` B) , ( 2nd ` B) >. e. A)
15		opelcnvg 4458	. . . . . 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 391	. . . . 5 |- (B e. A -> ( <.( 2nd ` B) , ( 1st ` B) >. e. `'A <-> <.( 1st ` B) , ( 2nd ` B) >. e. A))
17	16	ad2antrl 459	. . . 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 156	. . 3 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> <.( 2nd ` B) , ( 1st ` B) >. e. `'A)
19	12, 18	eqeltrd 2111	. 2 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> C e. `'A)
20		opswapg 4750	. . . . . 6 |- ((( 2nd ` B) e. _V /\ ( 1st ` B) e. _V) -> U. `' { <.( 2nd ` B) , ( 1st ` B) >. } = <.( 1st ` B) , ( 2nd ` B) >.)
21	8, 7, 20	syl2anc 391	. . . . 5 |- (B e. A -> U. `' { <.( 2nd ` B) , ( 1st ` B) >. } = <.( 1st ` B) , ( 2nd ` B) >.)
22	21	eqcomd 2042	. . . 4 |- (B e. A -> <.( 1st ` B) , ( 2nd ` B) >. = U. `' { <.( 2nd ` B) , ( 1st ` B) >. })
23	22	ad2antrl 459	. . 3 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> <.( 1st ` B) , ( 2nd ` B) >. = U. `' { <.( 2nd ` B) , ( 1st ` B) >. })
24	12	sneqd 3380	. . . . 5 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> { C } = { <.( 2nd ` B) , ( 1st ` B) >. })
25	24	cnveqd 4454	. . . 4 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> `' { C } = `' { <.( 2nd ` B) , ( 1st ` B) >. })
26	25	unieqd 3582	. . 3 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> U. `' { C } = U. `' { <.( 2nd ` B) , ( 1st ` B) >. })
27	23, 3, 26	3eqtr4d 2079	. 2 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> B = U. `' { C })
28	19, 27	jca 290	1 |- (( Rel A /\ (B e. A /\ C = U. `' {B })) -> ( C e. `'A /\ B = U. `' { C }))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   e. wcel 1390   _Vcvv 2551   {csn 3367   <.cop 3370   U.cuni 3571   `'ccnv 4287   Rel wrel 4293   ` cfv 4845   1stc1st 5707   2ndc2nd 5708

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710

This theorem is referenced by:  cnvf1o  5788