Theorem dissneqlem 37335

Description: This is the core of the proof of dissneq 37336, 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 22824	. . . 4 ⊢ (𝐵 ∈ (TopOn‘𝐴) → ({∅, 𝐴} ⊆ 𝐵 ∧ 𝐵 ⊆ 𝒫 𝐴))
2	1	adantl 481	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ({∅, 𝐴} ⊆ 𝐵 ∧ 𝐵 ⊆ 𝒫 𝐴))
3	2	simprd 495	. 2 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ⊆ 𝒫 𝐴)
4		velpw 4571	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
5		simp3 1138	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ∈ (TopOn‘𝐴))
6		df-ima 5654	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥)
7		resmpt 6011	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥) = (𝑧 ∈ 𝑥 ↦ {𝑧}))
8	7	rneqd 5905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ⊆ 𝐴 → ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥) = ran (𝑧 ∈ 𝑥 ↦ {𝑧}))
9	6, 8	eqtrid 2777	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ran (𝑧 ∈ 𝑥 ↦ {𝑧}))
10		rnmptsn 37330	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑧 ∈ 𝑥 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}
11	9, 10	eqtrdi 2781	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
12		imassrn 6045	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) ⊆ ran (𝑧 ∈ 𝐴 ↦ {𝑧})
13	11, 12	eqsstrrdi 3995	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ 𝐴 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ ran (𝑧 ∈ 𝐴 ↦ {𝑧}))
14		rnmptsn 37330	. . . . . . . . . . . . . . 15 ⊢ ran (𝑧 ∈ 𝐴 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
15	13, 14	sseqtrdi 3990	. . . . . . . . . . . . . 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 4602	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
18	17	eqeq2d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
19	18	cbvrexvw 3217	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧})
20	19	abbii 2797	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
21	16, 20	eqtri 2753	. . . . . . . . . . . . . 14 ⊢ 𝐶 = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
22	15, 21	sseqtrrdi 3991	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝐴 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
23	22	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
24		sstr 3958	. . . . . . . . . . . . . 14 ⊢ (({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
25	24	expcom 413	. . . . . . . . . . . . 13 ⊢ (𝐶 ⊆ 𝐵 → ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
26	25	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
27	23, 26	mpd 15	. . . . . . . . . . 11 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
28	27	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
29	5, 28	ssexd 5282	. . . . . . . . 9 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ∈ V)
30		isset 3464	. . . . . . . . 9 ⊢ ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ∈ V ↔ ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
31	29, 30	sylib 218	. . . . . . . 8 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
32		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐴 ↦ {𝑧}) = (𝑧 ∈ 𝐴 ↦ {𝑧})
33		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
34	32, 33	mptsnun 37334	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = ∪ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥))
35	11	unieqd 4887	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ 𝐴 → ∪ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
36	34, 35	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
37	36	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
38	27, 37	jca 511	. . . . . . . . . . 11 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵 ∧ 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}))
39		sseq1 3975	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑦 ⊆ 𝐵 ↔ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
40		unieq 4885	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ∪ 𝑦 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
41	40	eqeq2d 2741	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑥 = ∪ 𝑦 ↔ 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}))
42	39, 41	anbi12d 632	. . . . . . . . . . 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 247	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
44	43	eximdv 1917	. . . . . . . . 9 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
45	44	3adant3 1132	. . . . . . . 8 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
46	31, 45	mpd 15	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))
47	4, 46	syl3an2b 1406	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))
48	47	3com23 1126	. . . . 5 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴) ∧ 𝑥 ∈ 𝒫 𝐴) → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))
49	48	3expia 1121	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
50		topontop 22807	. . . . . . . 8 ⊢ (𝐵 ∈ (TopOn‘𝐴) → 𝐵 ∈ Top)
51		tgtop 22867	. . . . . . . 8 ⊢ (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵)
52	50, 51	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (topGen‘𝐵) = 𝐵)
53	52	eleq2d 2815	. . . . . 6 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ 𝐵))
54		eltg3 22856	. . . . . 6 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
55	53, 54	bitr3d 281	. . . . 5 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
56	55	adantl 481	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
57	49, 56	sylibrd 259	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵))
58	57	ssrdv 3955	. 2 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝒫 𝐴 ⊆ 𝐵)
59	3, 58	eqssd 3967	1 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592  {cpr 4594  ∪ cuni 4874   ↦ cmpt 5191  ran crn 5642   ↾ cres 5643   “ cima 5644  ‘cfv 6514  topGenctg 17407  Topctop 22787  TopOnctopon 22804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-topgen 17413  df-top 22788  df-topon 22805

This theorem is referenced by:  dissneq  37336