Theorem qliftrel 6385

Description: F, a function lift, is a subset of R X. S. (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 )

Assertion

Ref	Expression
qliftrel	|- ( ph -> F C_ ( ( X /. R ) X. Y ) )

Distinct variable groups:   ph, x   x, R   x, X   x, Y

Allowed substitution hints:   A( x)   F( x)

Proof of Theorem qliftrel

Step	Hyp	Ref	Expression
1		qlift.1	. 2 |- F = ran ( x e. X |-> <. [ x ] R , A >. )
2		qlift.2	. . 3 |- ( ( ph /\ x e. X ) -> A e. Y )
3		qlift.3	. . 3 |- ( ph -> R Er X )
4		qlift.4	. . 3 |- ( ph -> X e. _V )
5	1, 2, 3, 4	qliftlem 6384	. 2 |- ( ( ph /\ x e. X ) -> [ x ] R e. ( X /. R ) )
6	1, 5, 2	fliftrel 5585	1 |- ( ph -> F C_ ( ( X /. R ) X. Y ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   _Vcvv 2620   C_ wss 3000   <.cop 3453   |-> cmpt 3905   X. cxp 4449   ran crn 4452   Er wer 6303   [cec 6304   /.cqs 6305

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-er 6306  df-ec 6308  df-qs 6312

This theorem is referenced by: (None)