Theorem qliftfun 7282

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 7278	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6		eceq1 7234	. . 3 |- ( x = y -> [ x ] R = [ y ] R )
7		qliftfun.4	. . 3 |- ( x = y -> A = B )
8	1, 5, 2, 6, 7	fliftfun 6101	. 2 |- ( ph -> ( Fun F <-> A. x e. X A. y e. X ( [ x ] R = [ y ] R -> A = B ) ) )
9	3	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ x R y ) -> R Er X )
10		simpr 461	. . . . . . . . . . 11 |- ( ( ph /\ x R y ) -> x R y )
11	9, 10	ercl 7209	. . . . . . . . . 10 |- ( ( ph /\ x R y ) -> x e. X )
12	9, 10	ercl2 7211	. . . . . . . . . 10 |- ( ( ph /\ x R y ) -> y e. X )
13	11, 12	jca 532	. . . . . . . . 9 |- ( ( ph /\ x R y ) -> ( x e. X /\ y e. X ) )
14	13	ex 434	. . . . . . . 8 |- ( ph -> ( x R y -> ( x e. X /\ y e. X ) ) )
15	14	pm4.71rd 635	. . . . . . 7 |- ( ph -> ( x R y <-> ( ( x e. X /\ y e. X ) /\ x R y ) ) )
16	3	adantr 465	. . . . . . . . 9 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> R Er X )
17		simprl 755	. . . . . . . . 9 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> x e. X )
18	16, 17	erth 7242	. . . . . . . 8 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x R y <-> [ x ] R = [ y ] R ) )
19	18	pm5.32da 641	. . . . . . 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 253	. . . . . 6 |- ( ph -> ( x R y <-> ( ( x e. X /\ y e. X ) /\ [ x ] R = [ y ] R ) ) )
21	20	imbi1d 317	. . . . 5 |- ( ph -> ( ( x R y -> A = B ) <-> ( ( ( x e. X /\ y e. X ) /\ [ x ] R = [ y ] R ) -> A = B ) ) )
22		impexp 446	. . . . 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 261	. . . 4 |- ( ph -> ( ( x R y -> A = B ) <-> ( ( x e. X /\ y e. X ) -> ( [ x ] R = [ y ] R -> A = B ) ) ) )
24	23	2albidv 1682	. . 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 2853	. . 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 263	. 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 256	1 |- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1368   = wceq 1370   e. wcel 1758   A.wral 2793   _Vcvv 3065   <.cop 3978   class class class wbr 4387   |-> cmpt 4445   ran crn 4936   Fun wfun 5507   Er wer 7195   [cec 7196   /.cqs 7197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-er 7198  df-ec 7200  df-qs 7204

This theorem is referenced by:  qliftfund  7283  qliftfuns  7284