Theorem qliftfund 6677

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 115	. . 3 |- ( ph -> ( x R y -> A = B ) )
3	2	alrimivv 1889	. 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 6676	. 2 |- ( ph -> ( Fun F <-> A. x A. y ( x R y -> A = B ) ) )
10	3, 9	mpbird 167	1 |- ( ph -> Fun F )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   e. wcel 2167   _Vcvv 2763   <.cop 3625   class class class wbr 4033   |-> cmpt 4094   ran crn 4664   Fun wfun 5252   Er wer 6589   [cec 6590

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

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 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-er 6592  df-ec 6594  df-qs 6598

This theorem is referenced by: (None)