Theorem fliftfuns 5938

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 2374	. . . . 5 |- F/_ y <. A , B >.
3		nfcsb1v 3160	. . . . . 6 |- F/_ x [_ y / x ]_ A
4		nfcsb1v 3160	. . . . . 6 |- F/_ x [_ y / x ]_ B
5	3, 4	nfop 3878	. . . . 5 |- F/_ x <. [_ y / x ]_ A , [_ y / x ]_ B >.
6		csbeq1a 3136	. . . . . 6 |- ( x = y -> A = [_ y / x ]_ A )
7		csbeq1a 3136	. . . . . 6 |- ( x = y -> B = [_ y / x ]_ B )
8	6, 7	opeq12d 3870	. . . . 5 |- ( x = y -> <. A , B >. = <. [_ y / x ]_ A , [_ y / x ]_ B >. )
9	2, 5, 8	cbvmpt 4184	. . . 4 |- ( x e. X |-> <. A , B >. ) = ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
10	9	rneqi 4960	. . 3 |- ran ( x e. X |-> <. A , B >. ) = ran ( y e. X |-> <. [_ y / x ]_ A , [_ y / x ]_ B >. )
11	1, 10	eqtri 2252	. 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 2605	. . 3 |- ( ph -> A. x e. X A e. R )
14	3	nfel1 2385	. . . 4 |- F/ x [_ y / x ]_ A e. R
15	6	eleq1d 2300	. . . 4 |- ( x = y -> ( A e. R <-> [_ y / x ]_ A e. R ) )
16	14, 15	rspc 2904	. . 3 |- ( y e. X -> ( A. x e. X A e. R -> [_ y / x ]_ A e. R ) )
17	13, 16	mpan9 281	. 2 |- ( ( ph /\ y e. X ) -> [_ y / x ]_ A e. R )
18		flift.3	. . . 4 |- ( ( ph /\ x e. X ) -> B e. S )
19	18	ralrimiva 2605	. . 3 |- ( ph -> A. x e. X B e. S )
20	4	nfel1 2385	. . . 4 |- F/ x [_ y / x ]_ B e. S
21	7	eleq1d 2300	. . . 4 |- ( x = y -> ( B e. S <-> [_ y / x ]_ B e. S ) )
22	20, 21	rspc 2904	. . 3 |- ( y e. X -> ( A. x e. X B e. S -> [_ y / x ]_ B e. S ) )
23	19, 22	mpan9 281	. 2 |- ( ( ph /\ y e. X ) -> [_ y / x ]_ B e. S )
24		csbeq1 3130	. 2 |- ( y = z -> [_ y / x ]_ A = [_ z / x ]_ A )
25		csbeq1 3130	. 2 |- ( y = z -> [_ y / x ]_ B = [_ z / x ]_ B )
26	11, 17, 23, 24, 25	fliftfun 5936	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 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   [_csb 3127   <.cop 3672   |-> cmpt 4150   ran crn 4726   Fun wfun 5320

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334

This theorem is referenced by: (None)