Theorem dissneqlem 33187

Description: This is the core of the proof of dissneq 33188, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)

Hypothesis

Ref	Expression
dissneq.c	|- C = { u | E. x e. A u = { x } }

Assertion

Ref	Expression
dissneqlem	|- ( ( C C_ B /\ B e. (TopOn ` A ) ) -> B = ~P A )

Distinct variable groups:   u, A, x   x, B   x, C

Allowed substitution hints:   B( u)   C( u)

Proof of Theorem dissneqlem

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topgele 20734	. . . 4 |- ( B e. (TopOn ` A ) -> ( { (/) , A } C_ B /\ B C_ ~P A ) )
2	1	adantl 482	. . 3 |- ( ( C C_ B /\ B e. (TopOn ` A ) ) -> ( { (/) , A } C_ B /\ B C_ ~P A ) )
3	2	simprd 479	. 2 |- ( ( C C_ B /\ B e. (TopOn ` A ) ) -> B C_ ~P A )
4		selpw 4165	. . . . . . 7 |- ( x e. ~P A <-> x C_ A )
5		simp3 1063	. . . . . . . . . 10 |- ( ( C C_ B /\ x C_ A /\ B e. (TopOn ` A ) ) -> B e. (TopOn ` A ) )
6		df-ima 5127	. . . . . . . . . . . . . . . . . 18 |- ( ( z e. A |-> { z } ) " x ) = ran ( ( z e. A |-> { z } ) |` x )
7		resmpt 5449	. . . . . . . . . . . . . . . . . . 19 |- ( x C_ A -> ( ( z e. A |-> { z } ) |` x ) = ( z e. x |-> { z } ) )
8	7	rneqd 5353	. . . . . . . . . . . . . . . . . 18 |- ( x C_ A -> ran ( ( z e. A |-> { z } ) |` x ) = ran ( z e. x |-> { z } ) )
9	6, 8	syl5eq 2668	. . . . . . . . . . . . . . . . 17 |- ( x C_ A -> ( ( z e. A |-> { z } ) " x ) = ran ( z e. x |-> { z } ) )
10		rnmptsn 33182	. . . . . . . . . . . . . . . . 17 |- ran ( z e. x |-> { z } ) = { u | E. z e. x u = { z } }
11	9, 10	syl6eq 2672	. . . . . . . . . . . . . . . 16 |- ( x C_ A -> ( ( z e. A |-> { z } ) " x ) = { u | E. z e. x u = { z } } )
12		imassrn 5477	. . . . . . . . . . . . . . . 16 |- ( ( z e. A |-> { z } ) " x ) C_ ran ( z e. A |-> { z } )
13	11, 12	syl6eqssr 3656	. . . . . . . . . . . . . . 15 |- ( x C_ A -> { u | E. z e. x u = { z } } C_ ran ( z e. A |-> { z } ) )
14		rnmptsn 33182	. . . . . . . . . . . . . . 15 |- ran ( z e. A |-> { z } ) = { u | E. z e. A u = { z } }
15	13, 14	syl6sseq 3651	. . . . . . . . . . . . . 14 |- ( x C_ A -> { u | E. z e. x u = { z } } C_ { u | E. z e. A u = { z } } )
16		dissneq.c	. . . . . . . . . . . . . . 15 |- C = { u | E. x e. A u = { x } }
17		sneq 4187	. . . . . . . . . . . . . . . . . 18 |- ( x = z -> { x } = { z } )
18	17	eqeq2d 2632	. . . . . . . . . . . . . . . . 17 |- ( x = z -> ( u = { x } <-> u = { z } ) )
19	18	cbvrexv 3172	. . . . . . . . . . . . . . . 16 |- ( E. x e. A u = { x } <-> E. z e. A u = { z } )
20	19	abbii 2739	. . . . . . . . . . . . . . 15 |- { u | E. x e. A u = { x } } = { u | E. z e. A u = { z } }
21	16, 20	eqtri 2644	. . . . . . . . . . . . . 14 |- C = { u | E. z e. A u = { z } }
22	15, 21	syl6sseqr 3652	. . . . . . . . . . . . 13 |- ( x C_ A -> { u | E. z e. x u = { z } } C_ C )
23	22	adantl 482	. . . . . . . . . . . 12 |- ( ( C C_ B /\ x C_ A ) -> { u | E. z e. x u = { z } } C_ C )
24		sstr 3611	. . . . . . . . . . . . . 14 |- ( ( { u | E. z e. x u = { z } } C_ C /\ C C_ B ) -> { u | E. z e. x u = { z } } C_ B )
25	24	expcom 451	. . . . . . . . . . . . 13 |- ( C C_ B -> ( { u | E. z e. x u = { z } } C_ C -> { u | E. z e. x u = { z } } C_ B ) )
26	25	adantr 481	. . . . . . . . . . . 12 |- ( ( C C_ B /\ x C_ A ) -> ( { u | E. z e. x u = { z } } C_ C -> { u | E. z e. x u = { z } } C_ B ) )
27	23, 26	mpd 15	. . . . . . . . . . 11 |- ( ( C C_ B /\ x C_ A ) -> { u | E. z e. x u = { z } } C_ B )
28	27	3adant3 1081	. . . . . . . . . 10 |- ( ( C C_ B /\ x C_ A /\ B e. (TopOn ` A ) ) -> { u | E. z e. x u = { z } } C_ B )
29	5, 28	ssexd 4805	. . . . . . . . 9 |- ( ( C C_ B /\ x C_ A /\ B e. (TopOn ` A ) ) -> { u | E. z e. x u = { z } } e. _V )
30		isset 3207	. . . . . . . . 9 |- ( { u | E. z e. x u = { z } } e. _V <-> E. y y = { u | E. z e. x u = { z } } )
31	29, 30	sylib 208	. . . . . . . 8 |- ( ( C C_ B /\ x C_ A /\ B e. (TopOn ` A ) ) -> E. y y = { u | E. z e. x u = { z } } )
32		eqid 2622	. . . . . . . . . . . . . . 15 |- ( z e. A |-> { z } ) = ( z e. A |-> { z } )
33		eqid 2622	. . . . . . . . . . . . . . 15 |- { u | E. z e. A u = { z } } = { u | E. z e. A u = { z } }
34	32, 33	mptsnun 33186	. . . . . . . . . . . . . 14 |- ( x C_ A -> x = U. ( ( z e. A |-> { z } ) " x ) )
35	11	unieqd 4446	. . . . . . . . . . . . . 14 |- ( x C_ A -> U. ( ( z e. A |-> { z } ) " x ) = U. { u | E. z e. x u = { z } } )
36	34, 35	eqtrd 2656	. . . . . . . . . . . . 13 |- ( x C_ A -> x = U. { u | E. z e. x u = { z } } )
37	36	adantl 482	. . . . . . . . . . . 12 |- ( ( C C_ B /\ x C_ A ) -> x = U. { u | E. z e. x u = { z } } )
38	27, 37	jca 554	. . . . . . . . . . 11 |- ( ( C C_ B /\ x C_ A ) -> ( { u | E. z e. x u = { z } } C_ B /\ x = U. { u | E. z e. x u = { z } } ) )
39		sseq1 3626	. . . . . . . . . . . 12 |- ( y = { u | E. z e. x u = { z } } -> ( y C_ B <-> { u | E. z e. x u = { z } } C_ B ) )
40		unieq 4444	. . . . . . . . . . . . 13 |- ( y = { u | E. z e. x u = { z } } -> U. y = U. { u | E. z e. x u = { z } } )
41	40	eqeq2d 2632	. . . . . . . . . . . 12 |- ( y = { u | E. z e. x u = { z } } -> ( x = U. y <-> x = U. { u | E. z e. x u = { z } } ) )
42	39, 41	anbi12d 747	. . . . . . . . . . 11 |- ( y = { u | E. z e. x u = { z } } -> ( ( y C_ B /\ x = U. y ) <-> ( { u | E. z e. x u = { z } } C_ B /\ x = U. { u | E. z e. x u = { z } } ) ) )
43	38, 42	syl5ibrcom 237	. . . . . . . . . 10 |- ( ( C C_ B /\ x C_ A ) -> ( y = { u | E. z e. x u = { z } } -> ( y C_ B /\ x = U. y ) ) )
44	43	eximdv 1846	. . . . . . . . 9 |- ( ( C C_ B /\ x C_ A ) -> ( E. y y = { u | E. z e. x u = { z } } -> E. y ( y C_ B /\ x = U. y ) ) )
45	44	3adant3 1081	. . . . . . . 8 |- ( ( C C_ B /\ x C_ A /\ B e. (TopOn ` A ) ) -> ( E. y y = { u | E. z e. x u = { z } } -> E. y ( y C_ B /\ x = U. y ) ) )
46	31, 45	mpd 15	. . . . . . 7 |- ( ( C C_ B /\ x C_ A /\ B e. (TopOn ` A ) ) -> E. y ( y C_ B /\ x = U. y ) )
47	4, 46	syl3an2b 1363	. . . . . 6 |- ( ( C C_ B /\ x e. ~P A /\ B e. (TopOn ` A ) ) -> E. y ( y C_ B /\ x = U. y ) )
48	47	3com23 1271	. . . . 5 |- ( ( C C_ B /\ B e. (TopOn ` A ) /\ x e. ~P A ) -> E. y ( y C_ B /\ x = U. y ) )
49	48	3expia 1267	. . . 4 |- ( ( C C_ B /\ B e. (TopOn ` A ) ) -> ( x e. ~P A -> E. y ( y C_ B /\ x = U. y ) ) )
50		topontop 20718	. . . . . . . 8 |- ( B e. (TopOn ` A ) -> B e. Top )
51		tgtop 20777	. . . . . . . 8 |- ( B e. Top -> ( topGen ` B ) = B )
52	50, 51	syl 17	. . . . . . 7 |- ( B e. (TopOn ` A ) -> ( topGen ` B ) = B )
53	52	eleq2d 2687	. . . . . 6 |- ( B e. (TopOn ` A ) -> ( x e. ( topGen ` B ) <-> x e. B ) )
54		eltg3 20766	. . . . . 6 |- ( B e. (TopOn ` A ) -> ( x e. ( topGen ` B ) <-> E. y ( y C_ B /\ x = U. y ) ) )
55	53, 54	bitr3d 270	. . . . 5 |- ( B e. (TopOn ` A ) -> ( x e. B <-> E. y ( y C_ B /\ x = U. y ) ) )
56	55	adantl 482	. . . 4 |- ( ( C C_ B /\ B e. (TopOn ` A ) ) -> ( x e. B <-> E. y ( y C_ B /\ x = U. y ) ) )
57	49, 56	sylibrd 249	. . 3 |- ( ( C C_ B /\ B e. (TopOn ` A ) ) -> ( x e. ~P A -> x e. B ) )
58	57	ssrdv 3609	. 2 |- ( ( C C_ B /\ B e. (TopOn ` A ) ) -> ~P A C_ B )
59	3, 58	eqssd 3620	1 |- ( ( C C_ B /\ B e. (TopOn ` A ) ) -> B = ~P A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   U.cuni 4436   |-> cmpt 4729   ran crn 5115   |` cres 5116   "cima 5117   ` cfv 5888   topGenctg 16098   Topctop 20698  TopOnctopon 20715

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-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-fv 5896  df-topgen 16104  df-top 20699  df-topon 20716

This theorem is referenced by:  dissneq  33188