Theorem qliftfuns 6190

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 2178	. . . . 5 |- F/_ y <. [ x ] R , A >.
3		nfcv 2178	. . . . . 6 |- F/_ x [ y ] R
4		nfcsb1v 2882	. . . . . 6 |- F/_ x [_ y / x ]_ A
5	3, 4	nfop 3565	. . . . 5 |- F/_ x <. [ y ] R , [_ y / x ]_ A >.
6		eceq1 6141	. . . . . 6 |- ( x = y -> [ x ] R = [ y ] R )
7		csbeq1a 2860	. . . . . 6 |- ( x = y -> A = [_ y / x ]_ A )
8	6, 7	opeq12d 3557	. . . . 5 |- ( x = y -> <. [ x ] R , A >. = <. [ y ] R , [_ y / x ]_ A >. )
9	2, 5, 8	cbvmpt 3851	. . . 4 |- ( x e. X |-> <. [ x ] R , A >. ) = ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
10	9	rneqi 4562	. . 3 |- ran ( x e. X |-> <. [ x ] R , A >. ) = ran ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
11	1, 10	eqtri 2060	. 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 2392	. . 3 |- ( ph -> A. x e. X A e. Y )
14	4	nfel1 2188	. . . 4 |- F/ x [_ y / x ]_ A e. Y
15	7	eleq1d 2106	. . . 4 |- ( x = y -> ( A e. Y <-> [_ y / x ]_ A e. Y ) )
16	14, 15	rspc 2650	. . 3 |- ( y e. X -> ( A. x e. X A e. Y -> [_ y / x ]_ A e. Y ) )
17	13, 16	mpan9 265	. 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 2855	. 2 |- ( y = z -> [_ y / x ]_ A = [_ z / x ]_ A )
21	11, 17, 18, 19, 20	qliftfun 6188	1 |- ( ph -> ( Fun F <-> A. y A. z ( y R z -> [_ y / x ]_ A = [_ z / x ]_ A ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   A.wral 2306   _Vcvv 2557   [_csb 2852   <.cop 3378   class class class wbr 3764   |-> cmpt 3818   ran crn 4346   Fun wfun 4896   Er wer 6103   [cec 6104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-er 6106  df-ec 6108  df-qs 6112

This theorem is referenced by: (None)