Theorem fliftcnv 5919

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 2229	. . . . 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 5916	. . . 4 |- ( ph -> ran ( x e. X |-> <. B , A >. ) C_ ( S X. R ) )
5		relxp 4828	. . . 4 |- Rel ( S X. R )
6		relss 4806	. . . 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 1489	. . 3 |- ( ph -> Rel ran ( x e. X |-> <. B , A >. ) )
8		relcnv 5106	. . 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 5917	. . . . . 6 |- ( ph -> ( z F y <-> E. x e. X ( z = A /\ y = B ) ) )
12		vex 2802	. . . . . . 7 |- y e. _V
13		vex 2802	. . . . . . 7 |- z e. _V
14	12, 13	brcnv 4905	. . . . . 6 |- ( y `' F z <-> z F y )
15		ancom 266	. . . . . . 7 |- ( ( y = B /\ z = A ) <-> ( z = A /\ y = B ) )
16	15	rexbii 2537	. . . . . 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 5917	. . . . 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 4084	. . . 4 |- ( y `' F z <-> <. y , z >. e. `' F )
21		df-br 4084	. . . 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 4818	. 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 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   <.cop 3669   class class class wbr 4083   |-> cmpt 4145   X. cxp 4717   `'ccnv 4718   ran crn 4720   Rel wrel 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326

This theorem is referenced by: (None)