Theorem fnejoin2 32364

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 4452	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
2	1	eqcomd 2628	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 = ∪ {𝑋})
3	2	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → 𝑋 = ∪ {𝑋})
4		iftrue 4092	. . . . . . . . 9 ⊢ (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = {𝑋})
5	4	unieqd 4446	. . . . . . . 8 ⊢ (𝑆 = ∅ → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ {𝑋})
6	5	eqeq2d 2632	. . . . . . 7 ⊢ (𝑆 = ∅ → (𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ↔ 𝑋 = ∪ {𝑋}))
7	3, 6	syl5ibrcom 237	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 = ∅ → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
8		n0 3931	. . . . . . 7 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
9		unieq 4444	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
10	9	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑥))
11	10	rspccva 3308	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
12	11	3adant1 1079	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
13		fnejoin1 32363	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆))
14		eqid 2622	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 = ∪ 𝑥
15		eqid 2622	. . . . . . . . . . . 12 ⊢ ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)
16	14, 15	fnebas 32339	. . . . . . . . . . 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 2656	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
19	18	3expia 1267	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑥 ∈ 𝑆 → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
20	19	exlimdv 1861	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (∃𝑥 𝑥 ∈ 𝑆 → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
21	8, 20	syl5bi 232	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 ≠ ∅ → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
22	7, 21	pm2.61dne 2880	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
23		eqid 2622	. . . . . 6 ⊢ ∪ 𝑇 = ∪ 𝑇
24	15, 23	fnebas 32339	. . . . 5 ⊢ (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇)
25	22, 24	sylan9eq 2676	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇) → 𝑋 = ∪ 𝑇)
26	25	ex 450	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑋 = ∪ 𝑇))
27		fnetr 32346	. . . . . . 7 ⊢ ((𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆) ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇) → 𝑥Fne𝑇)
28	27	ex 450	. . . . . 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 1265	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑥 ∈ 𝑆) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑥Fne𝑇))
31	30	ralrimdva 2969	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → ∀𝑥 ∈ 𝑆 𝑥Fne𝑇))
32	26, 31	jcad 555	. 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 794	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 = ∪ 𝑇)
35	33, 34	eqtr3d 2658	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇)
36		sseq1 3626	. . . . 5 ⊢ ({𝑋} = if(𝑆 = ∅, {𝑋}, ∪ 𝑆) → ({𝑋} ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
37		sseq1 3626	. . . . 5 ⊢ (∪ 𝑆 = if(𝑆 = ∅, {𝑋}, ∪ 𝑆) → (∪ 𝑆 ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
38		elex 3212	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
39	38	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 ∈ V)
40	34, 39	eqeltrrd 2702	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ 𝑇 ∈ V)
41		uniexb 6973	. . . . . . . . . 10 ⊢ (𝑇 ∈ V ↔ ∪ 𝑇 ∈ V)
42	40, 41	sylibr 224	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
43		ssid 3624	. . . . . . . . 9 ⊢ 𝑇 ⊆ 𝑇
44		eltg3i 20765	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑇 ⊆ 𝑇) → ∪ 𝑇 ∈ (topGen‘𝑇))
45	42, 43, 44	sylancl 694	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ 𝑇 ∈ (topGen‘𝑇))
46	34, 45	eqeltrd 2701	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 ∈ (topGen‘𝑇))
47	46	snssd 4340	. . . . . 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 801	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)
50		fnetg 32340	. . . . . . . 8 ⊢ (𝑥Fne𝑇 → 𝑥 ⊆ (topGen‘𝑇))
51	50	ralimi 2952	. . . . . . 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 4469	. . . . . 6 ⊢ (∪ 𝑆 ⊆ (topGen‘𝑇) ↔ ∀𝑥 ∈ 𝑆 𝑥 ⊆ (topGen‘𝑇))
54	52, 53	sylibr 224	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∪ 𝑆 ⊆ (topGen‘𝑇))
55	36, 37, 48, 54	ifbothda 4123	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇))
56	15, 23	isfne4 32335	. . . 4 ⊢ (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇 ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
57	35, 55, 56	sylanbrc 698	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇)
58	57	ex 450	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → ((𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇))
59	32, 58	impbid 202	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177  ∪ cuni 4436   class class class wbr 4653  ‘cfv 5888  topGenctg 16098  Fnecfne 32331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104  df-fne 32332

This theorem is referenced by: (None)