Theorem fliftcnv 5842

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 2196	. . . . 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 5839	. . . 4 |- ( ph -> ran ( x e. X |-> <. B , A >. ) C_ ( S X. R ) )
5		relxp 4772	. . . 4 |- Rel ( S X. R )
6		relss 4750	. . . 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 1457	. . 3 |- ( ph -> Rel ran ( x e. X |-> <. B , A >. ) )
8		relcnv 5047	. . 3 |- Rel `' F
9	7, 8	jctil 312	. 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 5840	. . . . . 6 |- ( ph -> ( z F y <-> E. x e. X ( z = A /\ y = B ) ) )
12		vex 2766	. . . . . . 7 |- y e. _V
13		vex 2766	. . . . . . 7 |- z e. _V
14	12, 13	brcnv 4849	. . . . . 6 |- ( y `' F z <-> z F y )
15		ancom 266	. . . . . . 7 |- ( ( y = B /\ z = A ) <-> ( z = A /\ y = B ) )
16	15	rexbii 2504	. . . . . 6 |- ( E. x e. X ( y = B /\ z = A ) <-> E. x e. X ( z = A /\ y = B ) )
17	11, 14, 16	3bitr4g 223	. . . . 5 |- ( ph -> ( y `' F z <-> E. x e. X ( y = B /\ z = A ) ) )
18	1, 2, 3	fliftel 5840	. . . . 5 |- ( ph -> ( y ran ( x e. X |-> <. B , A >. ) z <-> E. x e. X ( y = B /\ z = A ) ) )
19	17, 18	bitr4d 191	. . . 4 |- ( ph -> ( y `' F z <-> y ran ( x e. X |-> <. B , A >. ) z ) )
20		df-br 4034	. . . 4 |- ( y `' F z <-> <. y , z >. e. `' F )
21		df-br 4034	. . . 4 |- ( y ran ( x e. X |-> <. B , A >. ) z <-> <. y , z >. e. ran ( x e. X |-> <. B , A >. ) )
22	19, 20, 21	3bitr3g 222	. . 3 |- ( ph -> ( <. y , z >. e. `' F <-> <. y , z >. e. ran ( x e. X |-> <. B , A >. ) ) )
23	22	eqrelrdv2 4762	. 2 |- ( ( ( Rel `' F /\ Rel ran ( x e. X |-> <. B , A >. ) ) /\ ph ) -> `' F = ran ( x e. X |-> <. B , A >. ) )
24	9, 23	mpancom 422	1 |- ( ph -> `' F = ran ( x e. X |-> <. B , A >. ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   E.wrex 2476   C_ wss 3157   <.cop 3625   class class class wbr 4033   |-> cmpt 4094   X. cxp 4661   `'ccnv 4662   ran crn 4664   Rel wrel 4668

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266

This theorem is referenced by: (None)