Theorem qliftfund 7833

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 450	. . 3 |- ( ph -> ( x R y -> A = B ) )
3	2	alrimivv 1856	. 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 7832	. 2 |- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) )
10	3, 9	mpbird 247	1 |- ( ph -> Fun F )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   Fun wfun 5882   Er wer 7739   [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-er 7742  df-ec 7744  df-qs 7748

This theorem is referenced by:  orbstafun  17744  frgpupf  18186