Theorem fnejoin2 33717

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 4857	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
2	1	eqcomd 2827	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 = ∪ {𝑋})
3	2	adantr 483	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → 𝑋 = ∪ {𝑋})
4		iftrue 4473	. . . . . . . . 9 ⊢ (𝑆 = ∅ → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = {𝑋})
5	4	unieqd 4852	. . . . . . . 8 ⊢ (𝑆 = ∅ → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ {𝑋})
6	5	eqeq2d 2832	. . . . . . 7 ⊢ (𝑆 = ∅ → (𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ↔ 𝑋 = ∪ {𝑋}))
7	3, 6	syl5ibrcom 249	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 = ∅ → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
8		n0 4310	. . . . . . 7 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
9		unieq 4849	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
10	9	eqeq2d 2832	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑥))
11	10	rspccva 3622	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
12	11	3adant1 1126	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
13		fnejoin1 33716	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆))
14		eqid 2821	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 = ∪ 𝑥
15		eqid 2821	. . . . . . . . . . . 12 ⊢ ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)
16	14, 15	fnebas 33692	. . . . . . . . . . 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 2856	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
19	18	3expia 1117	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑥 ∈ 𝑆 → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
20	19	exlimdv 1934	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (∃𝑥 𝑥 ∈ 𝑆 → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
21	8, 20	syl5bi 244	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 ≠ ∅ → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)))
22	7, 21	pm2.61dne 3103	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
23		eqid 2821	. . . . . 6 ⊢ ∪ 𝑇 = ∪ 𝑇
24	15, 23	fnebas 33692	. . . . 5 ⊢ (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇)
25	22, 24	sylan9eq 2876	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇) → 𝑋 = ∪ 𝑇)
26	25	ex 415	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑋 = ∪ 𝑇))
27		fnetr 33699	. . . . . . 7 ⊢ ((𝑥Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆) ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇) → 𝑥Fne𝑇)
28	27	ex 415	. . . . . 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 1114	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑥 ∈ 𝑆) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → 𝑥Fne𝑇))
31	30	ralrimdva 3189	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → ∀𝑥 ∈ 𝑆 𝑥Fne𝑇))
32	26, 31	jcad 515	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 → (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)))
33	22	adantr 483	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 = ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆))
34		simprl 769	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 = ∪ 𝑇)
35	33, 34	eqtr3d 2858	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇)
36		sseq1 3992	. . . . 5 ⊢ ({𝑋} = if(𝑆 = ∅, {𝑋}, ∪ 𝑆) → ({𝑋} ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
37		sseq1 3992	. . . . 5 ⊢ (∪ 𝑆 = if(𝑆 = ∅, {𝑋}, ∪ 𝑆) → (∪ 𝑆 ⊆ (topGen‘𝑇) ↔ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
38		elex 3512	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
39	38	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 ∈ V)
40	34, 39	eqeltrrd 2914	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ 𝑇 ∈ V)
41		uniexb 7486	. . . . . . . . . 10 ⊢ (𝑇 ∈ V ↔ ∪ 𝑇 ∈ V)
42	40, 41	sylibr 236	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑇 ∈ V)
43		ssid 3989	. . . . . . . . 9 ⊢ 𝑇 ⊆ 𝑇
44		eltg3i 21569	. . . . . . . . 9 ⊢ ((𝑇 ∈ V ∧ 𝑇 ⊆ 𝑇) → ∪ 𝑇 ∈ (topGen‘𝑇))
45	42, 43, 44	sylancl 588	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → ∪ 𝑇 ∈ (topGen‘𝑇))
46	34, 45	eqeltrd 2913	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → 𝑋 ∈ (topGen‘𝑇))
47	46	snssd 4742	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → {𝑋} ⊆ (topGen‘𝑇))
48	47	adantr 483	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ 𝑆 = ∅) → {𝑋} ⊆ (topGen‘𝑇))
49		simplrr 776	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)
50		fnetg 33693	. . . . . . . 8 ⊢ (𝑥Fne𝑇 → 𝑥 ⊆ (topGen‘𝑇))
51	50	ralimi 3160	. . . . . . 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 4870	. . . . . 6 ⊢ (∪ 𝑆 ⊆ (topGen‘𝑇) ↔ ∀𝑥 ∈ 𝑆 𝑥 ⊆ (topGen‘𝑇))
54	52, 53	sylibr 236	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) ∧ ¬ 𝑆 = ∅) → ∪ 𝑆 ⊆ (topGen‘𝑇))
55	36, 37, 48, 54	ifbothda 4504	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇))
56	15, 23	isfne4 33688	. . . 4 ⊢ (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (∪ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑇 ∧ if(𝑆 = ∅, {𝑋}, ∪ 𝑆) ⊆ (topGen‘𝑇)))
57	35, 55, 56	sylanbrc 585	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇)
58	57	ex 415	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → ((𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇))
59	32, 58	impbid 214	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (if(𝑆 = ∅, {𝑋}, ∪ 𝑆)Fne𝑇 ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑥Fne𝑇)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ⊆ wss 3936  ∅c0 4291  ifcif 4467  {csn 4567  ∪ cuni 4838   class class class wbr 5066  ‘cfv 6355  topGenctg 16711  Fnecfne 33684

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-topgen 16717  df-fne 33685

This theorem is referenced by: (None)