Theorem fnejoin1 34484

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 4868	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ⊆ ∪ 𝑆)
2	1	3ad2ant3 1133	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)
3	2	unissd 4846	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ⊆ ∪ ∪ 𝑆)
4		eqimss2 3974	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑦 → ∪ 𝑦 ⊆ 𝑋)
5		sspwuni 5025	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
6	4, 5	sylibr 233	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ⊆ 𝒫 𝑋)
7	6	ralimi 3086	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 → ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
8	7	3ad2ant2 1132	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
9		unissb 4870	. . . . . . 7 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
10	8, 9	sylibr 233	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝑆 ⊆ 𝒫 𝑋)
11		sspwuni 5025	. . . . . 6 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∪ ∪ 𝑆 ⊆ 𝑋)
12	10, 11	sylib 217	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ 𝑋)
13		unieq 4847	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
14	13	eqeq2d 2749	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐴))
15	14	rspccva 3551	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
16	15	3adant1 1128	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
17	12, 16	sseqtrd 3957	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ ∪ 𝐴)
18	3, 17	eqssd 3934	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 = ∪ ∪ 𝑆)
19		pwexg 5296	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
20	19	3ad2ant1 1131	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝒫 𝑋 ∈ V)
21	20, 10	ssexd 5243	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝑆 ∈ V)
22		bastg 22024	. . . . 5 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ⊆ (topGen‘∪ 𝑆))
23	21, 22	syl 17	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝑆 ⊆ (topGen‘∪ 𝑆))
24	2, 23	sstrd 3927	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ (topGen‘∪ 𝑆))
25		eqid 2738	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
26		eqid 2738	. . . 4 ⊢ ∪ ∪ 𝑆 = ∪ ∪ 𝑆
27	25, 26	isfne4 34456	. . 3 ⊢ (𝐴Fne∪ 𝑆 ↔ (∪ 𝐴 = ∪ ∪ 𝑆 ∧ 𝐴 ⊆ (topGen‘∪ 𝑆)))
28	18, 24, 27	sylanbrc 582	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fne∪ 𝑆)
29		ne0i 4265	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅)
30	29	3ad2ant3 1133	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑆 ≠ ∅)
31		ifnefalse 4468	. . 3 ⊢ (𝑆 ≠ ∅ → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑆)
32	30, 31	syl 17	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → if(𝑆 = ∅, {𝑋}, ∪ 𝑆) = ∪ 𝑆)
33	28, 32	breqtrrd 5098	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆))

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ifcif 4456  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836   class class class wbr 5070  ‘cfv 6418  topGenctg 17065  Fnecfne 34452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-topgen 17071  df-fne 34453

This theorem is referenced by:  fnejoin2  34485