Theorem fliftfuns 5766

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 2308	. . . . 5 |- F/_ y <. A , B >.
3		nfcsb1v 3078	. . . . . 6 |- F/_ x [_ y / x ]_ A
4		nfcsb1v 3078	. . . . . 6 |- F/_ x [_ y / x ]_ B
5	3, 4	nfop 3774	. . . . 5 |- F/_ x <. [_ y / x ]_ A , [_ y / x ]_ B >.
6		csbeq1a 3054	. . . . . 6 |- ( x = y -> A = [_ y / x ]_ A )
7		csbeq1a 3054	. . . . . 6 |- ( x = y -> B = [_ y / x ]_ B )
8	6, 7	opeq12d 3766	. . . . 5 |- ( x = y -> <. A , B >. = <. [_ y / x ]_ A , [_ y / x ]_ B >. )
9	2, 5, 8	cbvmpt 4077	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
10	9	rneqi 4832	. . 3 |- ran ( x e. X |-> <. A , B >. ) = ran ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
11	1, 10	eqtri 2186	. 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 2539	. . 3 |- ( ph -> A. x e. X A e. R )
14	3	nfel1 2319	. . . 4 |- F/ x [_ y / x ]_ A e. R
15	6	eleq1d 2235	. . . 4 |- ( x = y -> ( A e. R <-> [_ y / x ]_ A e. R ) )
16	14, 15	rspc 2824	. . 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 2539	. . 3 |- ( ph -> A. x e. X B e. S )
20	4	nfel1 2319	. . . 4 |- F/ x [_ y / x ]_ B e. S
21	7	eleq1d 2235	. . . 4 |- ( x = y -> ( B e. S <-> [_ y / x ]_ B e. S ) )
22	20, 21	rspc 2824	. . 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 3048	. 2 |- ( y = z -> [_ y / x ]_ A = [_ z / x ]_ A )
25		csbeq1 3048	. 2 |- ( y = z -> [_ y / x ]_ B = [_ z / x ]_ B )
26	11, 17, 23, 24, 25	fliftfun 5764	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 1343   e. wcel 2136   A.wral 2444   [_csb 3045   <.cop 3579   |-> cmpt 4043   ran crn 4605   Fun wfun 5182

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196

This theorem is referenced by: (None)