Theorem qliftfun 6829

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 6825	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6		eceq1 6780	. . 3 |- ( x = y -> [ x ] R = [ y ] R )
7		qliftfun.4	. . 3 |- ( x = y -> A = B )
8	1, 5, 2, 6, 7	fliftfun 5947	. 2 |- ( ph -> ( Fun F <-> A. x e. X A. y e. X ( [ x ] R = [ y ] R -> A = B ) ) )
9	3	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ x R y ) -> R Er X )
10		simpr 110	. . . . . . . . . . 11 |- ( ( ph /\ x R y ) -> x R y )
11	9, 10	ercl 6756	. . . . . . . . . 10 |- ( ( ph /\ x R y ) -> x e. X )
12	9, 10	ercl2 6758	. . . . . . . . . 10 |- ( ( ph /\ x R y ) -> y e. X )
13	11, 12	jca 306	. . . . . . . . 9 |- ( ( ph /\ x R y ) -> ( x e. X /\ y e. X ) )
14	13	ex 115	. . . . . . . 8 |- ( ph -> ( x R y -> ( x e. X /\ y e. X ) ) )
15	14	pm4.71rd 394	. . . . . . 7 |- ( ph -> ( x R y <-> ( ( x e. X /\ y e. X ) /\ x R y ) ) )
16	3	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> R Er X )
17		simprl 531	. . . . . . . . 9 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> x e. X )
18	16, 17	erth 6791	. . . . . . . 8 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> ( x R y <-> [ x ] R = [ y ] R ) )
19	18	pm5.32da 452	. . . . . . 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 188	. . . . . 6 |- ( ph -> ( x R y <-> ( ( x e. X /\ y e. X ) /\ [ x ] R = [ y ] R ) ) )
21	20	imbi1d 231	. . . . 5 |- ( ph -> ( ( x R y -> A = B ) <-> ( ( ( x e. X /\ y e. X ) /\ [ x ] R = [ y ] R ) -> A = B ) ) )
22		impexp 263	. . . . 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	bitrdi 196	. . . 4 |- ( ph -> ( ( x R y -> A = B ) <-> ( ( x e. X /\ y e. X ) -> ( [ x ] R = [ y ] R -> A = B ) ) ) )
24	23	2albidv 1915	. . 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 2552	. . 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	bitr4di 198	. 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 191	1 |- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1396   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   <.cop 3676   class class class wbr 4093   |-> cmpt 4155   ran crn 4732   Fun wfun 5327   Er wer 6742   [cec 6743   /.cqs 6744

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:  qliftfund  6830  qliftfuns  6831