Theorem fliftcnv 5455

Description: Converse of the relation F. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	|- F = ran ( x e. X |-> <. A , B >. )
flift.2	|- ( ( ph /\ x e. X ) -> A e. R )
flift.3	|- ( ( ph /\ x e. X ) -> B e. S )

Assertion

Ref	Expression
fliftcnv	|- ( ph -> `' F = ran ( x e. X |-> <. B , A >. ) )

Distinct variable groups:   x, R   ph, x   x, X   x, S

Allowed substitution hints:   A( x)   B( x)   F( x)

Proof of Theorem fliftcnv

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2081	. . . . 5 |- ran ( x e. X |-> <. B , A >. ) = ran ( x e. X |-> <. B , A >. )
2		flift.3	. . . . 5 |- ( ( ph /\ x e. X ) -> B e. S )
3		flift.2	. . . . 5 |- ( ( ph /\ x e. X ) -> A e. R )
4	1, 2, 3	fliftrel 5452	. . . 4 |- ( ph -> ran ( x e. X |-> <. B , A >. ) C_ ( S X. R ) )
5		relxp 4465	. . . 4 |- Rel ( S X. R )
6		relss 4445	. . . 4 |- ( ran ( x e. X |-> <. B , A >. ) C_ ( S X. R ) -> ( Rel ( S X. R ) -> Rel ran ( x e. X |-> <. B , A >. ) ) )
7	4, 5, 6	mpisyl 1375	. . 3 |- ( ph -> Rel ran ( x e. X |-> <. B , A >. ) )
8		relcnv 4723	. . 3 |- Rel `' F
9	7, 8	jctil 305	. 2 |- ( ph -> ( Rel `' F /\ Rel ran ( x e. X |-> <. B , A >. ) ) )
10		flift.1	. . . . . . 7 |- F = ran ( x e. X |-> <. A , B >. )
11	10, 3, 2	fliftel 5453	. . . . . 6 |- ( ph -> ( z F y <-> E. x e. X ( z = A /\ y = B ) ) )
12		vex 2604	. . . . . . 7 |- y e. _V
13		vex 2604	. . . . . . 7 |- z e. _V
14	12, 13	brcnv 4536	. . . . . 6 |- ( y `' F z <-> z F y )
15		ancom 262	. . . . . . 7 |- ( ( y = B /\ z = A ) <-> ( z = A /\ y = B ) )
16	15	rexbii 2373	. . . . . 6 |- ( E. x e. X ( y = B /\ z = A ) <-> E. x e. X ( z = A /\ y = B ) )
17	11, 14, 16	3bitr4g 221	. . . . 5 |- ( ph -> ( y `' F z <-> E. x e. X ( y = B /\ z = A ) ) )
18	1, 2, 3	fliftel 5453	. . . . 5 |- ( ph -> ( y ran ( x e. X |-> <. B , A >. ) z <-> E. x e. X ( y = B /\ z = A ) ) )
19	17, 18	bitr4d 189	. . . 4 |- ( ph -> ( y `' F z <-> y ran ( x e. X |-> <. B , A >. ) z ) )
20		df-br 3786	. . . 4 |- ( y `' F z <-> <. y , z >. e. `' F )
21		df-br 3786	. . . 4 |- ( y ran ( x e. X |-> <. B , A >. ) z <-> <. y , z >. e. ran ( x e. X |-> <. B , A >. ) )
22	19, 20, 21	3bitr3g 220	. . 3 |- ( ph -> ( <. y , z >. e. `' F <-> <. y , z >. e. ran ( x e. X |-> <. B , A >. ) ) )
23	22	eqrelrdv2 4457	. 2 |- ( ( ( Rel `' F /\ Rel ran ( x e. X |-> <. B , A >. ) ) /\ ph ) -> `' F = ran ( x e. X |-> <. B , A >. ) )
24	9, 23	mpancom 413	1 |- ( ph -> `' F = ran ( x e. X |-> <. B , A >. ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   E.wrex 2349   C_ wss 2973   <.cop 3401   class class class wbr 3785   |-> cmpt 3839   X. cxp 4361   `'ccnv 4362   ran crn 4364   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930

This theorem is referenced by: (None)