Theorem fnejoin1 36938

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

Assertion

Ref	Expression
fnejoin1	⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑆   𝑦,𝑋

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem fnejoin1

Step	Hyp	Ref	Expression
1		elssuni 4906	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ⊆ ∪ 𝑆)
2	1	3ad2ant3 1153	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)
3	2	unissd 4884	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ⊆ ∪ ∪ 𝑆)
4		eqimss2 3997	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑦 → ∪ 𝑦 ⊆ 𝑋)
5		sspwuni 5068	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
6	4, 5	sylibr 237	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ⊆ 𝒫 𝑋)
7	6	ralimi 3104	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 → ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
8	7	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
9		unissb 4908	. . . . . . 7 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
10	8, 9	sylibr 237	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝑆 ⊆ 𝒫 𝑋)
11		sspwuni 5068	. . . . . 6 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∪ ∪ 𝑆 ⊆ 𝑋)
12	10, 11	sylib 221	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ 𝑋)
13		unieq 4885	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
14	13	eqeq2d 2776	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐴))
15	14	rspccva 3582	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
16	15	3adant1 1148	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
17	12, 16	sseqtrd 3974	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ ∪ 𝐴)
18	3, 17	eqssd 3955	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 = ∪ ∪ 𝑆)
19		pwexg 5351	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
20	19	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝒫 𝑋 ∈ V)
21	20, 10	ssexd 5297	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝑆 ∈ V)
22		bastg 23175	. . . . 5 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ⊆ (topGen‘∪ 𝑆))
23	21, 22	syl 18	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝑆 ⊆ (topGen‘∪ 𝑆))
24	2, 23	sstrd 3948	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ (topGen‘∪ 𝑆))
25		eqid 2765	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
26		eqid 2765	. . . 4 ⊢ ∪ ∪ 𝑆 = ∪ ∪ 𝑆
27	25, 26	isfne4 36910	. . 3 ⊢ (𝐴Fne∪ 𝑆 ↔ (∪ 𝐴 = ∪ ∪ 𝑆 ∧ 𝐴 ⊆ (topGen‘∪ 𝑆)))
28	18, 24, 27	sylanbrc 595	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fne∪ 𝑆)
29		ne0i 4294	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅)
30	29	3ad2ant3 1153	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑆 ≠ ∅)
31		ifnefalse 4501	. . 3 ⊢ (𝑆 ≠ ∅ → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑆)
32	30, 31	syl 18	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑆)
33	28, 32	breqtrrd 5141	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  ifcif 4489  𝒫 cpw 4564  {csn 4591  ∪ cuni 4874   class class class wbr 5111  ‘cfv 6540  topGenctg 17514  Fnecfne 36906

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-topgen 17520  df-fne 36907

This theorem is used by:  fnejoin2  36939