Theorem fnejoin1 34243

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 4837	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ⊆ ∪ 𝑆)
2	1	3ad2ant3 1137	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)
3	2	unissd 4815	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ⊆ ∪ ∪ 𝑆)
4		eqimss2 3944	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑦 → ∪ 𝑦 ⊆ 𝑋)
5		sspwuni 4994	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
6	4, 5	sylibr 237	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ⊆ 𝒫 𝑋)
7	6	ralimi 3073	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 → ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
8	7	3ad2ant2 1136	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
9		unissb 4839	. . . . . . 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 4994	. . . . . 6 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∪ ∪ 𝑆 ⊆ 𝑋)
12	10, 11	sylib 221	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ 𝑋)
13		unieq 4816	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
14	13	eqeq2d 2747	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐴))
15	14	rspccva 3526	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
16	15	3adant1 1132	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
17	12, 16	sseqtrd 3927	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ ∪ 𝐴)
18	3, 17	eqssd 3904	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 = ∪ ∪ 𝑆)
19		pwexg 5256	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
20	19	3ad2ant1 1135	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝒫 𝑋 ∈ V)
21	20, 10	ssexd 5202	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝑆 ∈ V)
22		bastg 21817	. . . . 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 3897	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ (topGen‘∪ 𝑆))
25		eqid 2736	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
26		eqid 2736	. . . 4 ⊢ ∪ ∪ 𝑆 = ∪ ∪ 𝑆
27	25, 26	isfne4 34215	. . 3 ⊢ (𝐴Fne∪ 𝑆 ↔ (∪ 𝐴 = ∪ ∪ 𝑆 ∧ 𝐴 ⊆ (topGen‘∪ 𝑆)))
28	18, 24, 27	sylanbrc 586	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fne∪ 𝑆)
29		ne0i 4235	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅)
30	29	3ad2ant3 1137	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑆 ≠ ∅)
31		ifnefalse 4437	. . 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 5067	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆))

Syntax hints:   → wi 4   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  ∀wral 3051  Vcvv 3398   ⊆ wss 3853  ∅c0 4223  ifcif 4425  𝒫 cpw 4499  {csn 4527  ∪ cuni 4805   class class class wbr 5039  ‘cfv 6358  topGenctg 16896  Fnecfne 34211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6316  df-fun 6360  df-fv 6366  df-topgen 16902  df-fne 34212

This theorem is referenced by:  fnejoin2  34244