Theorem fliftfuns 5458

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 2219	. . . . 5 |- F/_ y <. A , B >.
3		nfcsb1v 2938	. . . . . 6 |- F/_ x [_ y / x ]_ A
4		nfcsb1v 2938	. . . . . 6 |- F/_ x [_ y / x ]_ B
5	3, 4	nfop 3586	. . . . 5 |- F/_ x <. [_ y / x ]_ A , [_ y / x ]_ B >.
6		csbeq1a 2916	. . . . . 6 |- ( x = y -> A = [_ y / x ]_ A )
7		csbeq1a 2916	. . . . . 6 |- ( x = y -> B = [_ y / x ]_ B )
8	6, 7	opeq12d 3578	. . . . 5 |- ( x = y -> <. A , B >. = <. [_ y / x ]_ A , [_ y / x ]_ B >. )
9	2, 5, 8	cbvmpt 3872	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
10	9	rneqi 4580	. . 3 |- ran ( x e. X |-> <. A , B >. ) = ran ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
11	1, 10	eqtri 2101	. 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 2434	. . 3 |- ( ph -> A. x e. X A e. R )
14	3	nfel1 2229	. . . 4 |- F/ x [_ y / x ]_ A e. R
15	6	eleq1d 2147	. . . 4 |- ( x = y -> ( A e. R <-> [_ y / x ]_ A e. R ) )
16	14, 15	rspc 2695	. . 3 |- ( y e. X -> ( A. x e. X A e. R -> [_ y / x ]_ A e. R ) )
17	13, 16	mpan9 275	. 2 |- ( ( ph /\ y e. X ) -> [_ y / x ]_ A e. R )
18		flift.3	. . . 4 |- ( ( ph /\ x e. X ) -> B e. S )
19	18	ralrimiva 2434	. . 3 |- ( ph -> A. x e. X B e. S )
20	4	nfel1 2229	. . . 4 |- F/ x [_ y / x ]_ B e. S
21	7	eleq1d 2147	. . . 4 |- ( x = y -> ( B e. S <-> [_ y / x ]_ B e. S ) )
22	20, 21	rspc 2695	. . 3 |- ( y e. X -> ( A. x e. X B e. S -> [_ y / x ]_ B e. S ) )
23	19, 22	mpan9 275	. 2 |- ( ( ph /\ y e. X ) -> [_ y / x ]_ B e. S )
24		csbeq1 2911	. 2 |- ( y = z -> [_ y / x ]_ A = [_ z / x ]_ A )
25		csbeq1 2911	. 2 |- ( y = z -> [_ y / x ]_ B = [_ z / x ]_ B )
26	11, 17, 23, 24, 25	fliftfun 5456	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 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   [_csb 2908   <.cop 3401   |-> cmpt 3839   ran crn 4364   Fun wfun 4916

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-csb 2909  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)