Theorem qliftfuns 6831

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 2375	. . . . 5 |- F/_ y <. [ x ] R , A >.
3		nfcv 2375	. . . . . 6 |- F/_ x [ y ] R
4		nfcsb1v 3161	. . . . . 6 |- F/_ x [_ y / x ]_ A
5	3, 4	nfop 3883	. . . . 5 |- F/_ x <. [ y ] R , [_ y / x ]_ A >.
6		eceq1 6780	. . . . . 6 |- ( x = y -> [ x ] R = [ y ] R )
7		csbeq1a 3137	. . . . . 6 |- ( x = y -> A = [_ y / x ]_ A )
8	6, 7	opeq12d 3875	. . . . 5 |- ( x = y -> <. [ x ] R , A >. = <. [ y ] R , [_ y / x ]_ A >. )
9	2, 5, 8	cbvmpt 4189	. . . 4 |- ( x e. X |-> <. [ x ] R , A >. ) = ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
10	9	rneqi 4966	. . 3 |- ran ( x e. X |-> <. [ x ] R , A >. ) = ran ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
11	1, 10	eqtri 2252	. 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 2606	. . 3 |- ( ph -> A. x e. X A e. Y )
14	4	nfel1 2386	. . . 4 |- F/ x [_ y / x ]_ A e. Y
15	7	eleq1d 2300	. . . 4 |- ( x = y -> ( A e. Y <-> [_ y / x ]_ A e. Y ) )
16	14, 15	rspc 2905	. . 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 3131	. 2 |- ( y = z -> [_ y / x ]_ A = [_ z / x ]_ A )
21	11, 17, 18, 19, 20	qliftfun 6829	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 1396   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   [_csb 3128   <.cop 3676   class class class wbr 4093   |-> cmpt 4155   ran crn 4732   Fun wfun 5327   Er wer 6742   [cec 6743

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-er 6745  df-ec 6747  df-qs 6751

This theorem is referenced by: (None)