Theorem qliftfuns 6585

Description: The function F is the unique function defined by F ` [ x ] = A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
qlift.1	|- F = ran ( x e. X |-> <. [ x ] R , A >. )
qlift.2	|- ( ( ph /\ x e. X ) -> A e. Y )
qlift.3	|- ( ph -> R Er X )
qlift.4	|- ( ph -> X e. _V )

Assertion

Ref	Expression
qliftfuns	|- ( ph -> ( Fun F <-> A. y A. z ( y R z -> [_ y / x ]_ A = [_ z / x ]_ A ) ) )

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

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

Proof of Theorem qliftfuns

Step	Hyp	Ref	Expression
1		qlift.1	. . 3 |- F = ran ( x e. X |-> <. [ x ] R , A >. )
2		nfcv 2308	. . . . 5 |- F/_ y <. [ x ] R , A >.
3		nfcv 2308	. . . . . 6 |- F/_ x [ y ] R
4		nfcsb1v 3078	. . . . . 6 |- F/_ x [_ y / x ]_ A
5	3, 4	nfop 3774	. . . . 5 |- F/_ x <. [ y ] R , [_ y / x ]_ A >.
6		eceq1 6536	. . . . . 6 |- ( x = y -> [ x ] R = [ y ] R )
7		csbeq1a 3054	. . . . . 6 |- ( x = y -> A = [_ y / x ]_ A )
8	6, 7	opeq12d 3766	. . . . 5 |- ( x = y -> <. [ x ] R , A >. = <. [ y ] R , [_ y / x ]_ A >. )
9	2, 5, 8	cbvmpt 4077	. . . 4 |- ( x e. X |-> <. [ x ] R , A >. ) = ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
10	9	rneqi 4832	. . 3 |- ran ( x e. X |-> <. [ x ] R , A >. ) = ran ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
11	1, 10	eqtri 2186	. 2 |- F = ran ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
12		qlift.2	. . . 4 |- ( ( ph /\ x e. X ) -> A e. Y )
13	12	ralrimiva 2539	. . 3 |- ( ph -> A. x e. X A e. Y )
14	4	nfel1 2319	. . . 4 |- F/ x [_ y / x ]_ A e. Y
15	7	eleq1d 2235	. . . 4 |- ( x = y -> ( A e. Y <-> [_ y / x ]_ A e. Y ) )
16	14, 15	rspc 2824	. . 3 |- ( y e. X -> ( A. x e. X A e. Y -> [_ y / x ]_ A e. Y ) )
17	13, 16	mpan9 279	. 2 |- ( ( ph /\ y e. X ) -> [_ y / x ]_ A e. Y )
18		qlift.3	. 2 |- ( ph -> R Er X )
19		qlift.4	. 2 |- ( ph -> X e. _V )
20		csbeq1 3048	. 2 |- ( y = z -> [_ y / x ]_ A = [_ z / x ]_ A )
21	11, 17, 18, 19, 20	qliftfun 6583	1 |- ( ph -> ( Fun F <-> A. y A. z ( y R z -> [_ y / x ]_ A = [_ z / x ]_ A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   [_csb 3045   <.cop 3579   class class class wbr 3982   |-> cmpt 4043   ran crn 4605   Fun wfun 5182   Er wer 6498   [cec 6499

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-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

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  df-er 6501  df-ec 6503  df-qs 6507

This theorem is referenced by: (None)