Theorem qliftfuns 6597

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 2312	. . . . 5 |- F/_ y <. [ x ] R , A >.
3		nfcv 2312	. . . . . 6 |- F/_ x [ y ] R
4		nfcsb1v 3082	. . . . . 6 |- F/_ x [_ y / x ]_ A
5	3, 4	nfop 3781	. . . . 5 |- F/_ x <. [ y ] R , [_ y / x ]_ A >.
6		eceq1 6548	. . . . . 6 |- ( x = y -> [ x ] R = [ y ] R )
7		csbeq1a 3058	. . . . . 6 |- ( x = y -> A = [_ y / x ]_ A )
8	6, 7	opeq12d 3773	. . . . 5 |- ( x = y -> <. [ x ] R , A >. = <. [ y ] R , [_ y / x ]_ A >. )
9	2, 5, 8	cbvmpt 4084	. . . 4 |- ( x e. X |-> <. [ x ] R , A >. ) = ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
10	9	rneqi 4839	. . 3 |- ran ( x e. X |-> <. [ x ] R , A >. ) = ran ( y e. X |-> <. [ y ] R , [_ y / x ]_ A >. )
11	1, 10	eqtri 2191	. 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 2543	. . 3 |- ( ph -> A. x e. X A e. Y )
14	4	nfel1 2323	. . . 4 |- F/ x [_ y / x ]_ A e. Y
15	7	eleq1d 2239	. . . 4 |- ( x = y -> ( A e. Y <-> [_ y / x ]_ A e. Y ) )
16	14, 15	rspc 2828	. . 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 3052	. 2 |- ( y = z -> [_ y / x ]_ A = [_ z / x ]_ A )
21	11, 17, 18, 19, 20	qliftfun 6595	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 1346   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   [_csb 3049   <.cop 3586   class class class wbr 3989   |-> cmpt 4050   ran crn 4612   Fun wfun 5192   Er wer 6510   [cec 6511

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-er 6513  df-ec 6515  df-qs 6519

This theorem is referenced by: (None)