Theorem qliftfun 6211

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 )
qliftfun.4	|- ( x = y -> A = B )

Assertion

Ref	Expression
qliftfun	|- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) )

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

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

Proof of Theorem qliftfun

Step	Hyp	Ref	Expression
1		qlift.1	. . 3 |- F = ran ( x e. X |-> <. [ x ] R , A >. )
2		qlift.2	. . . 4 |- ( ( ph /\ x e. X ) -> A e. Y )
3		qlift.3	. . . 4 |- ( ph -> R Er X )
4		qlift.4	. . . 4 |- ( ph -> X e. _V )
5	1, 2, 3, 4	qliftlem 6207	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6		eceq1 6164	. . 3 |- ( x = y -> [ x ] R = [ y ] R )
7		qliftfun.4	. . 3 |- ( x = y -> A = B )
8	1, 5, 2, 6, 7	fliftfun 5456	. 2 |- ( ph -> ( Fun F <-> A. x e. X A. y e. X ( [ x ] R = [ y ] R -> A = B ) ) )
9	3	adantr 270	. . . . . . . . . . 11 |- ( ( ph /\ x R y ) -> R Er X )
10		simpr 108	. . . . . . . . . . 11 |- ( ( ph /\ x R y ) -> x R y )
11	9, 10	ercl 6140	. . . . . . . . . 10 |- ( ( ph /\ x R y ) -> x e. X )
12	9, 10	ercl2 6142	. . . . . . . . . 10 |- ( ( ph /\ x R y ) -> y e. X )
13	11, 12	jca 300	. . . . . . . . 9 |- ( ( ph /\ x R y ) -> ( x e. X /\ y e. X ) )
14	13	ex 113	. . . . . . . 8 |- ( ph -> ( x R y -> ( x e. X /\ y e. X ) ) )
15	14	pm4.71rd 386	. . . . . . 7 |- ( ph -> ( x R y <-> ( ( x e. X /\ y e. X ) /\ x R y ) ) )
16	3	adantr 270	. . . . . . . . 9 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> R Er X )
17		simprl 497	. . . . . . . . 9 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> x e. X )
18	16, 17	erth 6173	. . . . . . . 8 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x R y <-> [ x ] R = [ y ] R ) )
19	18	pm5.32da 439	. . . . . . 7 |- ( ph -> ( ( ( x e. X /\ y e. X ) /\ x R y ) <-> ( ( x e. X /\ y e. X ) /\ [ x ] R = [ y ] R ) ) )
20	15, 19	bitrd 186	. . . . . 6 |- ( ph -> ( x R y <-> ( ( x e. X /\ y e. X ) /\ [ x ] R = [ y ] R ) ) )
21	20	imbi1d 229	. . . . 5 |- ( ph -> ( ( x R y -> A = B ) <-> ( ( ( x e. X /\ y e. X ) /\ [ x ] R = [ y ] R ) -> A = B ) ) )
22		impexp 259	. . . . 5 |- ( ( ( ( x e. X /\ y e. X ) /\ [ x ] R = [ y ] R ) -> A = B ) <-> ( ( x e. X /\ y e. X ) -> ( [ x ] R = [ y ] R -> A = B ) ) )
23	21, 22	syl6bb 194	. . . 4 |- ( ph -> ( ( x R y -> A = B ) <-> ( ( x e. X /\ y e. X ) -> ( [ x ] R = [ y ] R -> A = B ) ) ) )
24	23	2albidv 1788	. . 3 |- ( ph -> ( A. x A. y ( x R y -> A = B ) <-> A. x A. y ( ( x e. X /\ y e. X ) -> ( [ x ] R = [ y ] R -> A = B ) ) ) )
25		r2al 2385	. . 3 |- ( A. x e. X A. y e. X ( [ x ] R = [ y ] R -> A = B ) <-> A. x A. y ( ( x e. X /\ y e. X ) -> ( [ x ] R = [ y ] R -> A = B ) ) )
26	24, 25	syl6bbr 196	. 2 |- ( ph -> ( A. x A. y ( x R y -> A = B ) <-> A. x e. X A. y e. X ( [ x ] R = [ y ] R -> A = B ) ) )
27	8, 26	bitr4d 189	1 |- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   A.wral 2348   _Vcvv 2601   <.cop 3401   class class class wbr 3785   |-> cmpt 3839   ran crn 4364   Fun wfun 4916   Er wer 6126   [cec 6127   /.cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-er 6129  df-ec 6131  df-qs 6135

This theorem is referenced by:  qliftfund  6212  qliftfuns  6213