Theorem fliftfuns 5520

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 2225	. . . . 5 |- F/_ y <. A , B >.
3		nfcsb1v 2951	. . . . . 6 |- F/_ x [_ y / x ]_ A
4		nfcsb1v 2951	. . . . . 6 |- F/_ x [_ y / x ]_ B
5	3, 4	nfop 3615	. . . . 5 |- F/_ x <. [_ y / x ]_ A , [_ y / x ]_ B >.
6		csbeq1a 2929	. . . . . 6 |- ( x = y -> A = [_ y / x ]_ A )
7		csbeq1a 2929	. . . . . 6 |- ( x = y -> B = [_ y / x ]_ B )
8	6, 7	opeq12d 3607	. . . . 5 |- ( x = y -> <. A , B >. = <. [_ y / x ]_ A , [_ y / x ]_ B >. )
9	2, 5, 8	cbvmpt 3901	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
10	9	rneqi 4624	. . 3 |- ran ( x e. X |-> <. A , B >. ) = ran ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
11	1, 10	eqtri 2105	. 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 2442	. . 3 |- ( ph -> A. x e. X A e. R )
14	3	nfel1 2235	. . . 4 |- F/ x [_ y / x ]_ A e. R
15	6	eleq1d 2153	. . . 4 |- ( x = y -> ( A e. R <-> [_ y / x ]_ A e. R ) )
16	14, 15	rspc 2708	. . 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 2442	. . 3 |- ( ph -> A. x e. X B e. S )
20	4	nfel1 2235	. . . 4 |- F/ x [_ y / x ]_ B e. S
21	7	eleq1d 2153	. . . 4 |- ( x = y -> ( B e. S <-> [_ y / x ]_ B e. S ) )
22	20, 21	rspc 2708	. . 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 2924	. 2 |- ( y = z -> [_ y / x ]_ A = [_ z / x ]_ A )
25		csbeq1 2924	. 2 |- ( y = z -> [_ y / x ]_ B = [_ z / x ]_ B )
26	11, 17, 23, 24, 25	fliftfun 5518	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 1287   e. wcel 1436   A.wral 2355   [_csb 2921   <.cop 3428   |-> cmpt 3868   ran crn 4405   Fun wfun 4966

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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-fv 4980

This theorem is referenced by: (None)