Theorem fnejoin2 37137

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 4885	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
2	1	eqcomd 2767	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 = ∪ {𝑋})
3	2	adantr 486	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → 𝑋 = ∪ {𝑋})
4		iftrue 4488	. . . . . . . . 9 ⊢ (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = {𝑋})
5	4	unieqd 4880	. . . . . . . 8 ⊢ (𝑆 = ∅ → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ {𝑋})
6	5	eqeq2d 2772	. . . . . . 7 ⊢ (𝑆 = ∅ → (𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ↔ 𝑋 = ∪ {𝑋}))
7	3, 6	syl5ibrcom 250	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 = ∅ → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
8		n0 4300	. . . . . . 7 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
9		unieq 4878	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
10	9	eqeq2d 2772	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑥))
11	10	rspccva 3576	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
12	11	3adant1 1148	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
13		fnejoin1 37136	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆))
14		eqid 2761	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 = ∪ 𝑥
15		eqid 2761	. . . . . . . . . . . 12 ⊢ ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)
16	14, 15	fnebas 37112	. . . . . . . . . . 11 ⊢ (𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆) → ∪ 𝑥 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
17	13, 16	syl 18	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → ∪ 𝑥 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
18	12, 17	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
19	18	3expia 1139	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑥 ∈ 𝑆 → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
20	19	exlimdv 1966	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (∃𝑥 𝑥 ∈ 𝑆 → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
21	8, 20	biimtrid 245	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 ≠ ∅ → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
22	7, 21	pm2.61dne 3042	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
23		eqid 2761	. . . . . 6 ⊢ ∪ 𝑇 = ∪ 𝑇
24	15, 23	fnebas 37112	. . . . 5 ⊢ (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇)
25	22, 24	sylan9eq 2816	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇) → 𝑋 = ∪ 𝑇)
26	25	ex 418	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑋 = ∪ 𝑇))
27		fnetr 37119	. . . . . . 7 ⊢ ((𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆) ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇) → 𝑥Fne𝑇)
28	27	ex 418	. . . . . 6 ⊢ (𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑥Fne𝑇))
29	13, 28	syl 18	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑥Fne𝑇))
30	29	3expa 1136	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑥 ∈ 𝑆) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑥Fne𝑇))
31	30	ralrimdva 3163	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → ∀𝑥 ∈ 𝑆 𝑥Fne𝑇))
32	26, 31	jcad 522	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)))
33	22	adantr 486	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
34		simprl 783	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 = ∪ 𝑇)
35	33, 34	eqtr3d 2798	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇)
36		sseq1 3956	. . . . 5 ⊢ ({𝑋} = if(𝑆 = ∅, {𝑋}, ∪ 𝑆) → ({𝑋} ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
37		sseq1 3956	. . . . 5 ⊢ (∪ 𝑆 = if(𝑆 = ∅, {𝑋}, ∪ 𝑆) → (∪ 𝑆 ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
38		elex 3472	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
39	38	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 ∈ V)
40	34, 39	eqeltrrd 2862	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ 𝑇 ∈ V)
41		uniexb 7776	. . . . . . . . . 10 ⊢ (𝑇 ∈ V ↔ ∪ 𝑇 ∈ V)
42	40, 41	sylibr 237	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
43		ssid 3953	. . . . . . . . 9 ⊢ 𝑇 ⊆ 𝑇
44		eltg3i 23272	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑇 ⊆ 𝑇) → ∪ 𝑇 ∈ (topGen‘𝑇))
45	42, 43, 44	sylancl 598	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ 𝑇 ∈ (topGen‘𝑇))
46	34, 45	eqeltrd 2861	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 ∈ (topGen‘𝑇))
47	46	snssd 4747	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → {𝑋} ⊆ (topGen‘𝑇))
48	47	adantr 486	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ 𝑆 = ∅) → {𝑋} ⊆ (topGen‘𝑇))
49		simplrr 790	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)
50		fnetg 37113	. . . . . . . 8 ⊢ (𝑥Fne𝑇 → 𝑥 ⊆ (topGen‘𝑇))
51	50	ralimi 3100	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑆 𝑥Fne𝑇 → ∀𝑥 ∈ 𝑆 𝑥 ⊆ (topGen‘𝑇))
52	49, 51	syl 18	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑥 ⊆ (topGen‘𝑇))
53		unissb 4901	. . . . . 6 ⊢ (∪ 𝑆 ⊆ (topGen‘𝑇) ↔ ∀𝑥 ∈ 𝑆 𝑥 ⊆ (topGen‘𝑇))
54	52, 53	sylibr 237	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∪ 𝑆 ⊆ (topGen‘𝑇))
55	36, 37, 48, 54	ifbothda 4521	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇))
56	15, 23	isfne4 37108	. . . 4 ⊢ (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇 ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
57	35, 55, 56	sylanbrc 595	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇)
58	57	ex 418	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → ((𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇))
59	32, 58	impbid 215	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  ifcif 4482  {csn 4584  ∪ cuni 4867   class class class wbr 5103  ‘cfv 6537  topGenctg 17601  Fnecfne 37104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545  df-topgen 17607  df-fne 37105

This theorem is used by: (None)