Theorem isf34lem5 9488

Description: Lemma for isfin3-4 9492. (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 5694	. . . . . . 7 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		compss.a	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
3	2	isf34lem2 9483	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
4	3	adantr 473	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
5	4	frnd 6263	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ran 𝐹 ⊆ 𝒫 𝐴)
6	1, 5	syl5ss 3809	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ 𝑋) ⊆ 𝒫 𝐴)
7		simprl 788	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ 𝒫 𝐴)
8	4	fdmd 6265	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → dom 𝐹 = 𝒫 𝐴)
9	7, 8	sseqtr4d 3838	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ dom 𝐹)
10		sseqin2 4015	. . . . . . . . 9 ⊢ (𝑋 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑋) = 𝑋)
11	9, 10	sylib 210	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (dom 𝐹 ∩ 𝑋) = 𝑋)
12		simprr 790	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ≠ ∅)
13	11, 12	eqnetrd 3038	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (dom 𝐹 ∩ 𝑋) ≠ ∅)
14		imadisj 5701	. . . . . . . 8 ⊢ ((𝐹 “ 𝑋) = ∅ ↔ (dom 𝐹 ∩ 𝑋) = ∅)
15	14	necon3bii 3023	. . . . . . 7 ⊢ ((𝐹 “ 𝑋) ≠ ∅ ↔ (dom 𝐹 ∩ 𝑋) ≠ ∅)
16	13, 15	sylibr 226	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ 𝑋) ≠ ∅)
17	6, 16	jca 508	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ∧ (𝐹 “ 𝑋) ≠ ∅))
18	2	isf34lem4 9487	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ∧ (𝐹 “ 𝑋) ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ (𝐹 “ (𝐹 “ 𝑋)))
19	17, 18	syldan 586	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ (𝐹 “ (𝐹 “ 𝑋)))
20	2	isf34lem3 9485	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
21	20	adantrr 709	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
22	21	inteqd 4672	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∩ (𝐹 “ (𝐹 “ 𝑋)) = ∩ 𝑋)
23	19, 22	eqtrd 2833	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ 𝑋)
24	23	fveq2d 6415	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = (𝐹‘∩ 𝑋))
25	2	compsscnv 9481	. . . 4 ⊢ ◡𝐹 = 𝐹
26	25	fveq1i 6412	. . 3 ⊢ (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋)))
27	2	compssiso 9484	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴))
28		isof1o 6801	. . . . . 6 ⊢ (𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴) → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴)
29	27, 28	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴)
30	29	adantr 473	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴)
31		sspwuni 4802	. . . . . 6 ⊢ ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴)
32	6, 31	sylib 210	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∪ (𝐹 “ 𝑋) ⊆ 𝐴)
33		elpw2g 5019	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴))
34	33	adantr 473	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴))
35	32, 34	mpbird 249	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴)
36		f1ocnvfv1 6760	. . . 4 ⊢ ((𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴 ∧ ∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴) → (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
37	30, 35, 36	syl2anc 580	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
38	26, 37	syl5eqr 2847	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
39	24, 38	eqtr3d 2835	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∩ 𝑋) = ∪ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157   ≠ wne 2971   ∖ cdif 3766   ∩ cin 3768   ⊆ wss 3769  ∅c0 4115  𝒫 cpw 4349  ∪ cuni 4628  ∩ cint 4667   ↦ cmpt 4922  ^◡ccnv 5311  dom cdm 5312  ran crn 5313   “ cima 5315  ⟶wf 6097  –1-1-onto→wf1o 6100  ‘cfv 6101   Isom wiso 6102   [⊊] crpss 7170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-rpss 7171

This theorem is referenced by:  isf34lem7  9489