Theorem isf34lem5 10456

Description: Lemma for isfin3-4 10460. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Hypothesis

Ref	Expression
compss.a	⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))

Assertion

Ref	Expression
isf34lem5	⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∩ 𝑋) = ∪ (𝐹 “ 𝑋))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hints:   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem isf34lem5

Step	Hyp	Ref	Expression
1		imassrn 6197	. . . . . . 7 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		compss.a	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
3	2	isf34lem2 10451	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
4	3	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
5	4	frnd 6718	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ran 𝐹 ⊆ 𝒫 𝐴)
6	1, 5	sstrid 3919	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ 𝑋) ⊆ 𝒫 𝐴)
7		simprl 783	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ 𝒫 𝐴)
8	4	fdmd 6720	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → dom 𝐹 = 𝒫 𝐴)
9	7, 8	sseqtrrd 3945	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ dom 𝐹)
10		sseqin2 4120	. . . . . . . . 9 ⊢ (𝑋 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑋) = 𝑋)
11	9, 10	sylib 221	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (dom 𝐹 ∩ 𝑋) = 𝑋)
12		simprr 785	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ≠ ∅)
13	11, 12	eqnetrd 3022	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (dom 𝐹 ∩ 𝑋) ≠ ∅)
14		imadisj 6076	. . . . . . . 8 ⊢ ((𝐹 “ 𝑋) = ∅ ↔ (dom 𝐹 ∩ 𝑋) = ∅)
15	14	necon3bii 3007	. . . . . . 7 ⊢ ((𝐹 “ 𝑋) ≠ ∅ ↔ (dom 𝐹 ∩ 𝑋) ≠ ∅)
16	13, 15	sylibr 237	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ 𝑋) ≠ ∅)
17	6, 16	jca 521	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ∧ (𝐹 “ 𝑋) ≠ ∅))
18	2	isf34lem4 10455	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ∧ (𝐹 “ 𝑋) ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ (𝐹 “ (𝐹 “ 𝑋)))
19	17, 18	syldan 603	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ (𝐹 “ (𝐹 “ 𝑋)))
20	2	isf34lem3 10453	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
21	20	adantrr 730	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
22	21	inteqd 4911	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∩ (𝐹 “ (𝐹 “ 𝑋)) = ∩ 𝑋)
23	19, 22	eqtrd 2795	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ 𝑋)
24	23	fveq2d 6889	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = (𝐹‘∩ 𝑋))
25	2	compsscnv 10449	. . . 4 ⊢ ◡𝐹 = 𝐹
26	25	fveq1i 6886	. . 3 ⊢ (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋)))
27	2	compssiso 10452	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴))
28		isof1o 7331	. . . . 5 ⊢ (𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴) → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴)
29	27, 28	syl 18	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴)
30		sspwuni 5059	. . . . . 6 ⊢ ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴)
31	6, 30	sylib 221	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∪ (𝐹 “ 𝑋) ⊆ 𝐴)
32		elpw2g 5294	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴))
33	32	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴))
34	31, 33	mpbird 260	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴)
35		f1ocnvfv1 7284	. . . 4 ⊢ ((𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴 ∧ ∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴) → (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
36	29, 34, 35	syl2an2r 698	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
37	26, 36	eqtr3id 2809	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
38	24, 37	eqtr3d 2797	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∩ 𝑋) = ∪ (𝐹 “ 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278  𝒫 cpw 4556  ∪ cuni 4866  ∩ cint 4906   ↦ cmpt 5185  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   “ cima 5653  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539   [⊊] crpss 7738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-rpss 7739

This theorem is used by:  isf34lem7  10457