Theorem qliftfuns 6619

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 2319	. . . . 5 |- F/_ y <. [ x ] R , A >.
3		nfcv 2319	. . . . . 6 |- F/_ x [ y ] R
4		nfcsb1v 3091	. . . . . 6 |- F/_ x [_ y / x ]_ A
5	3, 4	nfop 3795	. . . . 5 |- F/_ x <. [ y ] R , [_ y / x ]_ A >.
6		eceq1 6570	. . . . . 6 |- ( x = y -> [ x ] R = [ y ] R )
7		csbeq1a 3067	. . . . . 6 |- ( x = y -> A = [_ y / x ]_ A )
8	6, 7	opeq12d 3787	. . . . 5 |- ( x = y -> <. [ x ] R , A >. = <. [ y ] R , [_ y / x ]_ A >. )
9	2, 5, 8	cbvmpt 4099	. . . 4 |- ( x e. X |-> <. [ x ] R , A >. ) = ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
10	9	rneqi 4856	. . 3 |- ran ( x e. X |-> <. [ x ] R , A >. ) = ran ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
11	1, 10	eqtri 2198	. 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 2550	. . 3 |- ( ph -> A. x e. X A e. Y )
14	4	nfel1 2330	. . . 4 |- F/ x [_ y / x ]_ A e. Y
15	7	eleq1d 2246	. . . 4 |- ( x = y -> ( A e. Y <-> [_ y / x ]_ A e. Y ) )
16	14, 15	rspc 2836	. . 3 |- ( y e. X -> ( A. x e. X A e. Y -> [_ y / x ]_ A e. Y ) )
17	13, 16	mpan9 281	. 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 3061	. 2 |- ( y = z -> [_ y / x ]_ A = [_ z / x ]_ A )
21	11, 17, 18, 19, 20	qliftfun 6617	1 |- ( ph -> ( Fun F <-> A. y A. z ( y R z -> [_ y / x ]_ A = [_ z / x ]_ A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2738   [_csb 3058   <.cop 3596   class class class wbr 4004   |-> cmpt 4065   ran crn 4628   Fun wfun 5211   Er wer 6532   [cec 6533

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-er 6535  df-ec 6537  df-qs 6541

This theorem is referenced by: (None)