Theorem qliftfund 6189

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 )
qliftfund.6	|- ( ( ph /\ x R y ) -> A = B )

Assertion

Ref	Expression
qliftfund	|- ( ph -> Fun F )

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 qliftfund

Step	Hyp	Ref	Expression
1		qliftfund.6	. . . 4 |- ( ( ph /\ x R y ) -> A = B )
2	1	ex 108	. . 3 |- ( ph -> ( x R y -> A = B ) )
3	2	alrimivv 1755	. 2 |- ( ph -> A. x A. y ( x R y -> A = B ) )
4		qlift.1	. . 3 |- F = ran ( x e. X |-> <. [ x ] R , A >. )
5		qlift.2	. . 3 |- ( ( ph /\ x e. X ) -> A e. Y )
6		qlift.3	. . 3 |- ( ph -> R Er X )
7		qlift.4	. . 3 |- ( ph -> X e. _V )
8		qliftfun.4	. . 3 |- ( x = y -> A = B )
9	4, 5, 6, 7, 8	qliftfun 6188	. 2 |- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) )
10	3, 9	mpbird 156	1 |- ( ph -> Fun F )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   e. wcel 1393   _Vcvv 2557   <.cop 3378   class class class wbr 3764   |-> cmpt 3818   ran crn 4346   Fun wfun 4896   Er wer 6103   [cec 6104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-er 6106  df-ec 6108  df-qs 6112

This theorem is referenced by: (None)