Theorem qliftfun 6685

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 6681	. . 3 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6		eceq1 6636	. . 3 |- ( x = y -> [ x ] R = [ y ] R )
7		qliftfun.4	. . 3 |- ( x = y -> A = B )
8	1, 5, 2, 6, 7	fliftfun 5846	. 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 6612	. . . . . . . . . 10 |- ( ( ph /\ x R y ) -> x e. X )
12	9, 10	ercl2 6614	. . . . . . . . . 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 529	. . . . . . . . 9 |- ( ( ph /\ ( x e. X /\ y e. X ) ) -> x e. X )
18	16, 17	erth 6647	. . . . . . . 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 1881	. . 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 2516	. . 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 1362   = wceq 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   <.cop 3626   class class class wbr 4034   |-> cmpt 4095   ran crn 4665   Fun wfun 5253   Er wer 6598   [cec 6599   /.cqs 6600

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-er 6601  df-ec 6603  df-qs 6607

This theorem is referenced by:  qliftfund  6686  qliftfuns  6687