Theorem dissneqlem 37702

Description: This is the core of the proof of dissneq 37703, 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 22913	. . . 4 ⊢ (𝐵 ∈ (TopOn‘𝐴) → ({∅, 𝐴} ⊆ 𝐵 ∧ 𝐵 ⊆ 𝒫 𝐴))
2	1	adantl 482	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ({∅, 𝐴} ⊆ 𝐵 ∧ 𝐵 ⊆ 𝒫 𝐴))
3	2	simprd 496	. 2 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ⊆ 𝒫 𝐴)
4		velpw 4534	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
5		simp3 1144	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ∈ (TopOn‘𝐴))
6		df-ima 5631	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥)
7		resmpt 5989	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥) = (𝑧 ∈ 𝑥 ↦ {𝑧}))
8	7	rneqd 5880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ⊆ 𝐴 → ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥) = ran (𝑧 ∈ 𝑥 ↦ {𝑧}))
9	6, 8	eqtrid 2786	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ran (𝑧 ∈ 𝑥 ↦ {𝑧}))
10		rnmptsn 37697	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑧 ∈ 𝑥 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}
11	9, 10	eqtrdi 2790	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
12		imassrn 6023	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) ⊆ ran (𝑧 ∈ 𝐴 ↦ {𝑧})
13	11, 12	eqsstrrdi 3960	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ 𝐴 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ ran (𝑧 ∈ 𝐴 ↦ {𝑧}))
14		rnmptsn 37697	. . . . . . . . . . . . . . 15 ⊢ ran (𝑧 ∈ 𝐴 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
15	13, 14	sseqtrdi 3955	. . . . . . . . . . . . . 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 4565	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
18	17	eqeq2d 2750	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
19	18	cbvrexvw 3218	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧})
20	19	abbii 2806	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
21	16, 20	eqtri 2762	. . . . . . . . . . . . . 14 ⊢ 𝐶 = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
22	15, 21	sseqtrrdi 3956	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝐴 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
23	22	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶)
24		sstr 3923	. . . . . . . . . . . . . 14 ⊢ (({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
25	24	expcom 414	. . . . . . . . . . . . 13 ⊢ (𝐶 ⊆ 𝐵 → ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐶 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
26	25	adantr 481	. . . . . . . . . . . 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 1138	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
29	5, 28	ssexd 5252	. . . . . . . . 9 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ∈ V)
30		isset 3445	. . . . . . . . 9 ⊢ ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ∈ V ↔ ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
31	29, 30	sylib 219	. . . . . . . 8 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
32		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐴 ↦ {𝑧}) = (𝑧 ∈ 𝐴 ↦ {𝑧})
33		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
34	32, 33	mptsnun 37701	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = ∪ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥))
35	11	unieqd 4851	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ 𝐴 → ∪ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
36	34, 35	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
37	36	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
38	27, 37	jca 516	. . . . . . . . . . 11 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → ({𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵 ∧ 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}))
39		sseq1 3940	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑦 ⊆ 𝐵 ↔ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
40		unieq 4849	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ∪ 𝑦 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
41	40	eqeq2d 2750	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑥 = ∪ 𝑦 ↔ 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}))
42	39, 41	anbi12d 638	. . . . . . . . . . 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 248	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
44	43	eximdv 1924	. . . . . . . . 9 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
45	44	3adant3 1138	. . . . . . . 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 1412	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))
48	47	3com23 1132	. . . . 5 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴) ∧ 𝑥 ∈ 𝒫 𝐴) → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))
49	48	3expia 1127	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
50		topontop 22896	. . . . . . . 8 ⊢ (𝐵 ∈ (TopOn‘𝐴) → 𝐵 ∈ Top)
51		tgtop 22956	. . . . . . . 8 ⊢ (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵)
52	50, 51	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (topGen‘𝐵) = 𝐵)
53	52	eleq2d 2825	. . . . . 6 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ 𝐵))
54		eltg3 22945	. . . . . 6 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
55	53, 54	bitr3d 282	. . . . 5 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
56	55	adantl 482	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
57	49, 56	sylibrd 260	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵))
58	57	ssrdv 3921	. 2 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝒫 𝐴 ⊆ 𝐵)
59	3, 58	eqssd 3932	1 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  {csn 4555  {cpr 4557  ∪ cuni 4838   ↦ cmpt 5153  ran crn 5619   ↾ cres 5620   “ cima 5621  ‘cfv 6485  topGenctg 17391  Topctop 22876  TopOnctopon 22893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fv 6493  df-topgen 17397  df-top 22877  df-topon 22894

This theorem is referenced by:  dissneq  37703