Theorem fnejoin2 34697

Description: Join of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
fnejoin2	⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)))

Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝑉   𝑥,𝑋,𝑦   𝑥,𝑇

Allowed substitution hints:   𝑇(𝑦)   𝑉(𝑦)

Proof of Theorem fnejoin2

Step	Hyp	Ref	Expression
1		unisng 4874	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
2	1	eqcomd 2742	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 = ∪ {𝑋})
3	2	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → 𝑋 = ∪ {𝑋})
4		iftrue 4480	. . . . . . . . 9 ⊢ (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = {𝑋})
5	4	unieqd 4867	. . . . . . . 8 ⊢ (𝑆 = ∅ → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ {𝑋})
6	5	eqeq2d 2747	. . . . . . 7 ⊢ (𝑆 = ∅ → (𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ↔ 𝑋 = ∪ {𝑋}))
7	3, 6	syl5ibrcom 246	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 = ∅ → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
8		n0 4294	. . . . . . 7 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
9		unieq 4864	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
10	9	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑥))
11	10	rspccva 3569	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
12	11	3adant1 1129	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
13		fnejoin1 34696	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆))
14		eqid 2736	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 = ∪ 𝑥
15		eqid 2736	. . . . . . . . . . . 12 ⊢ ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)
16	14, 15	fnebas 34672	. . . . . . . . . . 11 ⊢ (𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆) → ∪ 𝑥 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
17	13, 16	syl 17	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → ∪ 𝑥 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
18	12, 17	eqtrd 2776	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
19	18	3expia 1120	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑥 ∈ 𝑆 → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
20	19	exlimdv 1935	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (∃𝑥 𝑥 ∈ 𝑆 → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
21	8, 20	biimtrid 241	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 ≠ ∅ → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
22	7, 21	pm2.61dne 3028	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
23		eqid 2736	. . . . . 6 ⊢ ∪ 𝑇 = ∪ 𝑇
24	15, 23	fnebas 34672	. . . . 5 ⊢ (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇)
25	22, 24	sylan9eq 2796	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇) → 𝑋 = ∪ 𝑇)
26	25	ex 413	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑋 = ∪ 𝑇))
27		fnetr 34679	. . . . . . 7 ⊢ ((𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆) ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇) → 𝑥Fne𝑇)
28	27	ex 413	. . . . . 6 ⊢ (𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑥Fne𝑇))
29	13, 28	syl 17	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑥Fne𝑇))
30	29	3expa 1117	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑥 ∈ 𝑆) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑥Fne𝑇))
31	30	ralrimdva 3147	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → ∀𝑥 ∈ 𝑆 𝑥Fne𝑇))
32	26, 31	jcad 513	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)))
33	22	adantr 481	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
34		simprl 768	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 = ∪ 𝑇)
35	33, 34	eqtr3d 2778	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇)
36		sseq1 3957	. . . . 5 ⊢ ({𝑋} = if(𝑆 = ∅, {𝑋}, ∪ 𝑆) → ({𝑋} ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
37		sseq1 3957	. . . . 5 ⊢ (∪ 𝑆 = if(𝑆 = ∅, {𝑋}, ∪ 𝑆) → (∪ 𝑆 ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
38		elex 3459	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
39	38	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 ∈ V)
40	34, 39	eqeltrrd 2838	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ 𝑇 ∈ V)
41		uniexb 7677	. . . . . . . . . 10 ⊢ (𝑇 ∈ V ↔ ∪ 𝑇 ∈ V)
42	40, 41	sylibr 233	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
43		ssid 3954	. . . . . . . . 9 ⊢ 𝑇 ⊆ 𝑇
44		eltg3i 22218	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑇 ⊆ 𝑇) → ∪ 𝑇 ∈ (topGen‘𝑇))
45	42, 43, 44	sylancl 586	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ 𝑇 ∈ (topGen‘𝑇))
46	34, 45	eqeltrd 2837	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 ∈ (topGen‘𝑇))
47	46	snssd 4757	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → {𝑋} ⊆ (topGen‘𝑇))
48	47	adantr 481	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ 𝑆 = ∅) → {𝑋} ⊆ (topGen‘𝑇))
49		simplrr 775	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)
50		fnetg 34673	. . . . . . . 8 ⊢ (𝑥Fne𝑇 → 𝑥 ⊆ (topGen‘𝑇))
51	50	ralimi 3082	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑆 𝑥Fne𝑇 → ∀𝑥 ∈ 𝑆 𝑥 ⊆ (topGen‘𝑇))
52	49, 51	syl 17	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑥 ⊆ (topGen‘𝑇))
53		unissb 4888	. . . . . 6 ⊢ (∪ 𝑆 ⊆ (topGen‘𝑇) ↔ ∀𝑥 ∈ 𝑆 𝑥 ⊆ (topGen‘𝑇))
54	52, 53	sylibr 233	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∪ 𝑆 ⊆ (topGen‘𝑇))
55	36, 37, 48, 54	ifbothda 4512	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇))
56	15, 23	isfne4 34668	. . . 4 ⊢ (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇 ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
57	35, 55, 56	sylanbrc 583	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇)
58	57	ex 413	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → ((𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇))
59	32, 58	impbid 211	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2940  ∀wral 3061  Vcvv 3441   ⊆ wss 3898  ∅c0 4270  ifcif 4474  {csn 4574  ∪ cuni 4853   class class class wbr 5093  ‘cfv 6480  topGenctg 17246  Fnecfne 34664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6432  df-fun 6482  df-fv 6488  df-topgen 17252  df-fne 34665

This theorem is referenced by: (None)