Theorem fnejoin1 36767

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 4908	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ⊆ ∪ 𝑆)
2	1	3ad2ant3 1151	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆)
3	2	unissd 4886	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ⊆ ∪ ∪ 𝑆)
4		eqimss2 4004	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑦 → ∪ 𝑦 ⊆ 𝑋)
5		sspwuni 5070	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋)
6	4, 5	sylibr 237	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑦 → 𝑦 ⊆ 𝒫 𝑋)
7	6	ralimi 3108	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 → ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
8	7	3ad2ant2 1150	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 𝑦 ⊆ 𝒫 𝑋)
9		unissb 4910	. . . . . . 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 5070	. . . . . 6 ⊢ (∪ 𝑆 ⊆ 𝒫 𝑋 ↔ ∪ ∪ 𝑆 ⊆ 𝑋)
12	10, 11	sylib 221	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ 𝑋)
13		unieq 4887	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
14	13	eqeq2d 2780	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐴))
15	14	rspccva 3589	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
16	15	3adant1 1146	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
17	12, 16	sseqtrd 3981	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∪ 𝑆 ⊆ ∪ 𝐴)
18	3, 17	eqssd 3962	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 = ∪ ∪ 𝑆)
19		pwexg 5350	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
20	19	3ad2ant1 1149	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝒫 𝑋 ∈ V)
21	20, 10	ssexd 5295	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝑆 ∈ V)
22		bastg 23091	. . . . 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 3955	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ (topGen‘∪ 𝑆))
25		eqid 2769	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
26		eqid 2769	. . . 4 ⊢ ∪ ∪ 𝑆 = ∪ ∪ 𝑆
27	25, 26	isfne4 36739	. . 3 ⊢ (𝐴Fne∪ 𝑆 ↔ (∪ 𝐴 = ∪ ∪ 𝑆 ∧ 𝐴 ⊆ (topGen‘∪ 𝑆)))
28	18, 24, 27	sylanbrc 594	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fne∪ 𝑆)
29		ne0i 4302	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅)
30	29	3ad2ant3 1151	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑆 ≠ ∅)
31		ifnefalse 4504	. . 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 5143	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴Fneif(𝑆 = ∅, {𝑋}, ∪ 𝑆))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ifcif 4492  𝒫 cpw 4567  {csn 4594  ∪ cuni 4876   class class class wbr 5113  ‘cfv 6537  topGenctg 17489  Fnecfne 36735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-topgen 17495  df-fne 36736

This theorem is referenced by:  fnejoin2  36768