Theorem fliftfuns 5692

Description: The function F is the unique function defined by F ` A = B, provided that the well-definedness condition holds. (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
fliftfuns	|- ( ph -> ( Fun F <-> A. y e. X A. z e. X ( [_ y / x ]_ A = [_ z / x ]_ A -> [_ y / x ]_ B = [_ z / x ]_ B ) ) )

Distinct variable groups:   y, z, A   y, B, z   x, z, y, R   y, F, z   ph, x, y, z   x, X, y, z   x, S, y, z

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

Proof of Theorem fliftfuns

Step	Hyp	Ref	Expression
1		flift.1	. . 3 |- F = ran ( x e. X |-> <. A , B >. )
2		nfcv 2279	. . . . 5 |- F/_ y <. A , B >.
3		nfcsb1v 3030	. . . . . 6 |- F/_ x [_ y / x ]_ A
4		nfcsb1v 3030	. . . . . 6 |- F/_ x [_ y / x ]_ B
5	3, 4	nfop 3716	. . . . 5 |- F/_ x <. [_ y / x ]_ A , [_ y / x ]_ B >.
6		csbeq1a 3007	. . . . . 6 |- ( x = y -> A = [_ y / x ]_ A )
7		csbeq1a 3007	. . . . . 6 |- ( x = y -> B = [_ y / x ]_ B )
8	6, 7	opeq12d 3708	. . . . 5 |- ( x = y -> <. A , B >. = <. [_ y / x ]_ A , [_ y / x ]_ B >. )
9	2, 5, 8	cbvmpt 4018	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
10	9	rneqi 4762	. . 3 |- ran ( x e. X |-> <. A , B >. ) = ran ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
11	1, 10	eqtri 2158	. 2 |- F = ran ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
12		flift.2	. . . 4 |- ( ( ph /\ x e. X ) -> A e. R )
13	12	ralrimiva 2503	. . 3 |- ( ph -> A. x e. X A e. R )
14	3	nfel1 2290	. . . 4 |- F/ x [_ y / x ]_ A e. R
15	6	eleq1d 2206	. . . 4 |- ( x = y -> ( A e. R <-> [_ y / x ]_ A e. R ) )
16	14, 15	rspc 2778	. . 3 |- ( y e. X -> ( A. x e. X A e. R -> [_ y / x ]_ A e. R ) )
17	13, 16	mpan9 279	. 2 |- ( ( ph /\ y e. X ) -> [_ y / x ]_ A e. R )
18		flift.3	. . . 4 |- ( ( ph /\ x e. X ) -> B e. S )
19	18	ralrimiva 2503	. . 3 |- ( ph -> A. x e. X B e. S )
20	4	nfel1 2290	. . . 4 |- F/ x [_ y / x ]_ B e. S
21	7	eleq1d 2206	. . . 4 |- ( x = y -> ( B e. S <-> [_ y / x ]_ B e. S ) )
22	20, 21	rspc 2778	. . 3 |- ( y e. X -> ( A. x e. X B e. S -> [_ y / x ]_ B e. S ) )
23	19, 22	mpan9 279	. 2 |- ( ( ph /\ y e. X ) -> [_ y / x ]_ B e. S )
24		csbeq1 3001	. 2 |- ( y = z -> [_ y / x ]_ A = [_ z / x ]_ A )
25		csbeq1 3001	. 2 |- ( y = z -> [_ y / x ]_ B = [_ z / x ]_ B )
26	11, 17, 23, 24, 25	fliftfun 5690	1 |- ( ph -> ( Fun F <-> A. y e. X A. z e. X ( [_ y / x ]_ A = [_ z / x ]_ A -> [_ y / x ]_ B = [_ z / x ]_ B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2414   [_csb 2998   <.cop 3525   |-> cmpt 3984   ran crn 4535   Fun wfun 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126

This theorem is referenced by: (None)