Theorem qtopcmplem 21510

Description: Lemma for qtopcmp 21511 and qtopconn 21512. (Contributed by Mario Carneiro, 24-Mar-2015.)

Hypotheses

Ref	Expression
qtopcmp.1	|- X = U. J
qtopcmplem.1	|- ( J e. A -> J e. Top )
qtopcmplem.2	|- ( ( J e. A /\ F : X -onto-> U. ( J qTop F ) /\ F e. ( J Cn ( J qTop F ) ) ) -> ( J qTop F ) e. A )

Assertion

Ref	Expression
qtopcmplem	|- ( ( J e. A /\ F Fn X ) -> ( J qTop F ) e. A )

Proof of Theorem qtopcmplem

Step	Hyp	Ref	Expression
1		simpl 473	. 2 |- ( ( J e. A /\ F Fn X ) -> J e. A )
2		simpr 477	. . . 4 |- ( ( J e. A /\ F Fn X ) -> F Fn X )
3		dffn4 6121	. . . 4 |- ( F Fn X <-> F : X -onto-> ran F )
4	2, 3	sylib 208	. . 3 |- ( ( J e. A /\ F Fn X ) -> F : X -onto-> ran F )
5		qtopcmplem.1	. . . . . 6 |- ( J e. A -> J e. Top )
6		qtopcmp.1	. . . . . . 7 |- X = U. J
7	6	qtopuni 21505	. . . . . 6 |- ( ( J e. Top /\ F : X -onto-> ran F ) -> ran F = U. ( J qTop F ) )
8	5, 7	sylan 488	. . . . 5 |- ( ( J e. A /\ F : X -onto-> ran F ) -> ran F = U. ( J qTop F ) )
9	3, 8	sylan2b 492	. . . 4 |- ( ( J e. A /\ F Fn X ) -> ran F = U. ( J qTop F ) )
10		foeq3 6113	. . . 4 |- ( ran F = U. ( J qTop F ) -> ( F : X -onto-> ran F <-> F : X -onto-> U. ( J qTop F ) ) )
11	9, 10	syl 17	. . 3 |- ( ( J e. A /\ F Fn X ) -> ( F : X -onto-> ran F <-> F : X -onto-> U. ( J qTop F ) ) )
12	4, 11	mpbid 222	. 2 |- ( ( J e. A /\ F Fn X ) -> F : X -onto-> U. ( J qTop F ) )
13	6	toptopon 20722	. . . 4 |- ( J e. Top <-> J e. (TopOn ` X ) )
14	5, 13	sylib 208	. . 3 |- ( J e. A -> J e. (TopOn ` X ) )
15		qtopid 21508	. . 3 |- ( ( J e. (TopOn ` X ) /\ F Fn X ) -> F e. ( J Cn ( J qTop F ) ) )
16	14, 15	sylan 488	. 2 |- ( ( J e. A /\ F Fn X ) -> F e. ( J Cn ( J qTop F ) ) )
17		qtopcmplem.2	. 2 |- ( ( J e. A /\ F : X -onto-> U. ( J qTop F ) /\ F e. ( J Cn ( J qTop F ) ) ) -> ( J qTop F ) e. A )
18	1, 12, 16, 17	syl3anc 1326	1 |- ( ( J e. A /\ F Fn X ) -> ( J qTop F ) e. A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   U.cuni 4436   ran crn 5115   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   qTop cqtop 16163   Topctop 20698  TopOnctopon 20715   Cn ccn 21028

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-rep 4771  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-reu 2919  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-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-qtop 16167  df-top 20699  df-topon 20716  df-cn 21031

This theorem is referenced by:  qtopcmp  21511  qtopconn  21512  qtoppconn  31218