Theorem dissneqlem 37967

Description: This is the core of the proof of dissneq 37968, 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 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}

Assertion

Ref	Expression
dissneqlem	⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (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 23068	. . . 4 ⊢ (𝐵 ∈ (TopOn‘𝐴) → ({∅, 𝐴} ⊆ 𝐵 ∧ 𝐵 ⊆ 𝒫 𝐴))
2	1	adantl 486	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ({∅, 𝐴} ⊆ 𝐵 ∧ 𝐵 ⊆ 𝒫 𝐴))
3	2	simprd 500	. 2 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ⊆ 𝒫 𝐴)
4		velpw 4568	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
5		simp3 1156	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ∈ (TopOn‘𝐴))
6		df-ima 5676	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥)
7		resmpt 6041	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥) = (𝑧 ∈ 𝑥 ↦ {𝑧}))
8	7	rneqd 5930	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ⊆ 𝐴 → ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥) = ran (𝑧 ∈ 𝑥 ↦ {𝑧}))
9	6, 8	eqtrid 2810	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ran (𝑧 ∈ 𝑥 ↦ {𝑧}))
10		rnmptsn 37962	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑧 ∈ 𝑥 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}
11	9, 10	eqtrdi 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
12		imassrn 6075	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) ⊆ ran (𝑧 ∈ 𝐴 ↦ {𝑧})
13	11, 12	eqsstrrdi 3983	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ 𝐴 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ ran (𝑧 ∈ 𝐴 ↦ {𝑧}))
14		rnmptsn 37962	. . . . . . . . . . . . . . 15 ⊢ ran (𝑧 ∈ 𝐴 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
15	13, 14	sseqtrdi 3978	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ 𝐴 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}})
16		dissneq.c	. . . . . . . . . . . . . . 15 ⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
17		sneq 4600	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
18	17	eqeq2d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
19	18	cbvrexvw 3244	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧})
20	19	abbii 2830	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
21	16, 20	eqtri 2786	. . . . . . . . . . . . . 14 ⊢ 𝐶 = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
22	15, 21	sseqtrrdi 3979	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝐴 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
23	22	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
24		sstr 3946	. . . . . . . . . . . . . 14 ⊢ (({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
25	24	expcom 418	. . . . . . . . . . . . 13 ⊢ (𝐶 ⊆ 𝐵 → ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
26	25	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
27	23, 26	mpd 16	. . . . . . . . . . 11 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
28	27	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
29	5, 28	ssexd 5296	. . . . . . . . 9 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ∈ V)
30		isset 3469	. . . . . . . . 9 ⊢ ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ∈ V ↔ ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
31	29, 30	sylib 221	. . . . . . . 8 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
32		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐴 ↦ {𝑧}) = (𝑧 ∈ 𝐴 ↦ {𝑧})
33		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
34	32, 33	mptsnun 37966	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = ∪ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥))
35	11	unieqd 4886	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ 𝐴 → ∪ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
36	34, 35	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
37	36	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
38	27, 37	jca 520	. . . . . . . . . . 11 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵 ∧ 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}))
39		sseq1 3963	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑦 ⊆ 𝐵 ↔ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
40		unieq 4884	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ∪ 𝑦 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
41	40	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑥 = ∪ 𝑦 ↔ 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}))
42	39, 41	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ((𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦) ↔ ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵 ∧ 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})))
43	38, 42	syl5ibrcom 250	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
44	43	eximdv 1947	. . . . . . . . 9 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
45	44	3adant3 1150	. . . . . . . 8 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
46	31, 45	mpd 16	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))
47	4, 46	syl3an2b 1431	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))
48	47	3com23 1144	. . . . 5 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴) ∧ 𝑥 ∈ 𝒫 𝐴) → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))
49	48	3expia 1139	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
50		topontop 23051	. . . . . . . 8 ⊢ (𝐵 ∈ (TopOn‘𝐴) → 𝐵 ∈ Top)
51		tgtop 23111	. . . . . . . 8 ⊢ (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵)
52	50, 51	syl 18	. . . . . . 7 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (topGen‘𝐵) = 𝐵)
53	52	eleq2d 2849	. . . . . 6 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ 𝐵))
54		eltg3 23100	. . . . . 6 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
55	53, 54	bitr3d 284	. . . . 5 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
56	55	adantl 486	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
57	49, 56	sylibrd 262	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵))
58	57	ssrdv 3944	. 2 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝒫 𝐴 ⊆ 𝐵)
59	3, 58	eqssd 3955	1 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∃wrex 3089  Vcvv 3455   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  {csn 4590  {cpr 4592  ∪ cuni 4873   ↦ cmpt 5193  ran crn 5664   ↾ cres 5665   “ cima 5666  ‘cfv 6538  topGenctg 17491  Topctop 23031  TopOnctopon 23048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fv 6546  df-topgen 17497  df-top 23032  df-topon 23049

This theorem is referenced by:  dissneq  37968