Theorem fnejoin1 36336

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 4961	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ⊆ ∪ 𝑆)
2	1	3ad2ant3 1135	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)
3	2	unissd 4941	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ⊆ ∪ ∪ 𝑆)
4		eqimss2 4068	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑦 → ∪ 𝑦 ⊆ 𝑋)
5		sspwuni 5123	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
6	4, 5	sylibr 234	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ⊆ 𝒫 𝑋)
7	6	ralimi 3089	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 → ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
8	7	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
9		unissb 4963	. . . . . . 7 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
10	8, 9	sylibr 234	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝑆 ⊆ 𝒫 𝑋)
11		sspwuni 5123	. . . . . 6 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∪ ∪ 𝑆 ⊆ 𝑋)
12	10, 11	sylib 218	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ 𝑋)
13		unieq 4942	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
14	13	eqeq2d 2751	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐴))
15	14	rspccva 3634	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
16	15	3adant1 1130	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
17	12, 16	sseqtrd 4049	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ ∪ 𝐴)
18	3, 17	eqssd 4026	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 = ∪ ∪ 𝑆)
19		pwexg 5396	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
20	19	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝒫 𝑋 ∈ V)
21	20, 10	ssexd 5342	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝑆 ∈ V)
22		bastg 22996	. . . . 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 4019	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ (topGen‘∪ 𝑆))
25		eqid 2740	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
26		eqid 2740	. . . 4 ⊢ ∪ ∪ 𝑆 = ∪ ∪ 𝑆
27	25, 26	isfne4 36308	. . 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 4364	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅)
30	29	3ad2ant3 1135	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑆 ≠ ∅)
31		ifnefalse 4560	. . 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 5194	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  ifcif 4548  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931   class class class wbr 5166  ‘cfv 6575  topGenctg 17499  Fnecfne 36304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6527  df-fun 6577  df-fv 6583  df-topgen 17505  df-fne 36305

This theorem is referenced by:  fnejoin2  36337