Theorem dissneqlem 37670

Description: This is the core of the proof of dissneq 37671, 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 22905	. . . 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 4547	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
5		simp3 1139	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 ∈ (TopOn‘𝐴))
6		df-ima 5637	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥)
7		resmpt 5996	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥) = (𝑧 ∈ 𝑥 ↦ {𝑧}))
8	7	rneqd 5887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ⊆ 𝐴 → ran ((𝑧 ∈ 𝐴 ↦ {𝑧}) ↾ 𝑥) = ran (𝑧 ∈ 𝑥 ↦ {𝑧}))
9	6, 8	eqtrid 2784	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ran (𝑧 ∈ 𝑥 ↦ {𝑧}))
10		rnmptsn 37665	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑧 ∈ 𝑥 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}
11	9, 10	eqtrdi 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝐴 → ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
12		imassrn 6030	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) ⊆ ran (𝑧 ∈ 𝐴 ↦ {𝑧})
13	11, 12	eqsstrrdi 3968	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ 𝐴 → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ ran (𝑧 ∈ 𝐴 ↦ {𝑧}))
14		rnmptsn 37665	. . . . . . . . . . . . . . 15 ⊢ ran (𝑧 ∈ 𝐴 ↦ {𝑧}) = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
15	13, 14	sseqtrdi 3963	. . . . . . . . . . . . . 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 4578	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
18	17	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑢 = {𝑥} ↔ 𝑢 = {𝑧}))
19	18	cbvrexvw 3217	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧})
20	19	abbii 2804	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
21	16, 20	eqtri 2760	. . . . . . . . . . . . . 14 ⊢ 𝐶 = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
22	15, 21	sseqtrrdi 3964	. . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . 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 1133	. . . . . . . . . 10 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵)
29	5, 28	ssexd 5261	. . . . . . . . 9 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ∈ V)
30		isset 3444	. . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝐴 ↦ {𝑧}) = (𝑧 ∈ 𝐴 ↦ {𝑧})
33		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}} = {𝑢 ∣ ∃𝑧 ∈ 𝐴 𝑢 = {𝑧}}
34	32, 33	mptsnun 37669	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ 𝐴 → 𝑥 = ∪ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥))
35	11	unieqd 4864	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ 𝐴 → ∪ ((𝑧 ∈ 𝐴 ↦ {𝑧}) “ 𝑥) = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
36	34, 35	eqtrd 2772	. . . . . . . . . . . . 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 3948	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑦 ⊆ 𝐵 ↔ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} ⊆ 𝐵))
40		unieq 4862	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ∪ 𝑦 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}})
41	40	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → (𝑥 = ∪ 𝑦 ↔ 𝑥 = ∪ {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}}))
42	39, 41	anbi12d 633	. . . . . . . . . . 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 1919	. . . . . . . . 9 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴) → (∃𝑦 𝑦 = {𝑢 ∣ ∃𝑧 ∈ 𝑥 𝑢 = {𝑧}} → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
45	44	3adant3 1133	. . . . . . . 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 1407	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴 ∧ 𝐵 ∈ (TopOn‘𝐴)) → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))
48	47	3com23 1127	. . . . 5 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴) ∧ 𝑥 ∈ 𝒫 𝐴) → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦))
49	48	3expia 1122	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → (𝑥 ∈ 𝒫 𝐴 → ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)))
50		topontop 22888	. . . . . . . 8 ⊢ (𝐵 ∈ (TopOn‘𝐴) → 𝐵 ∈ Top)
51		tgtop 22948	. . . . . . . 8 ⊢ (𝐵 ∈ Top → (topGen‘𝐵) = 𝐵)
52	50, 51	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (topGen‘𝐵) = 𝐵)
53	52	eleq2d 2823	. . . . . 6 ⊢ (𝐵 ∈ (TopOn‘𝐴) → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ 𝐵))
54		eltg3 22937	. . . . . 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 3928	. 2 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝒫 𝐴 ⊆ 𝐵)
59	3, 58	eqssd 3940	1 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  {csn 4568  {cpr 4570  ∪ cuni 4851   ↦ cmpt 5167  ran crn 5625   ↾ cres 5626   “ cima 5627  ‘cfv 6492  topGenctg 17391  Topctop 22868  TopOnctopon 22885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-topgen 17397  df-top 22869  df-topon 22886

This theorem is referenced by:  dissneq  37671