Theorem isf34lem5 10290

Description: Lemma for isfin3-4 10294. (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 6030	. . . . . . 7 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		compss.a	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
3	2	isf34lem2 10285	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
5	4	frnd 6670	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ran 𝐹 ⊆ 𝒫 𝐴)
6	1, 5	sstrid 3945	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ 𝑋) ⊆ 𝒫 𝐴)
7		simprl 770	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ 𝒫 𝐴)
8	4	fdmd 6672	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → dom 𝐹 = 𝒫 𝐴)
9	7, 8	sseqtrrd 3971	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ dom 𝐹)
10		sseqin2 4175	. . . . . . . . 9 ⊢ (𝑋 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑋) = 𝑋)
11	9, 10	sylib 218	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (dom 𝐹 ∩ 𝑋) = 𝑋)
12		simprr 772	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ≠ ∅)
13	11, 12	eqnetrd 2999	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (dom 𝐹 ∩ 𝑋) ≠ ∅)
14		imadisj 6039	. . . . . . . 8 ⊢ ((𝐹 “ 𝑋) = ∅ ↔ (dom 𝐹 ∩ 𝑋) = ∅)
15	14	necon3bii 2984	. . . . . . 7 ⊢ ((𝐹 “ 𝑋) ≠ ∅ ↔ (dom 𝐹 ∩ 𝑋) ≠ ∅)
16	13, 15	sylibr 234	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ 𝑋) ≠ ∅)
17	6, 16	jca 511	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ∧ (𝐹 “ 𝑋) ≠ ∅))
18	2	isf34lem4 10289	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ∧ (𝐹 “ 𝑋) ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ (𝐹 “ (𝐹 “ 𝑋)))
19	17, 18	syldan 591	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ (𝐹 “ (𝐹 “ 𝑋)))
20	2	isf34lem3 10287	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
21	20	adantrr 717	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
22	21	inteqd 4907	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∩ (𝐹 “ (𝐹 “ 𝑋)) = ∩ 𝑋)
23	19, 22	eqtrd 2771	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ 𝑋)
24	23	fveq2d 6838	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = (𝐹‘∩ 𝑋))
25	2	compsscnv 10283	. . . 4 ⊢ ◡𝐹 = 𝐹
26	25	fveq1i 6835	. . 3 ⊢ (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋)))
27	2	compssiso 10286	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴))
28		isof1o 7269	. . . . 5 ⊢ (𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴) → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴)
29	27, 28	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴)
30		sspwuni 5055	. . . . . 6 ⊢ ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴)
31	6, 30	sylib 218	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∪ (𝐹 “ 𝑋) ⊆ 𝐴)
32		elpw2g 5278	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴))
33	32	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴))
34	31, 33	mpbird 257	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴)
35		f1ocnvfv1 7222	. . . 4 ⊢ ((𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴 ∧ ∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴) → (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
36	29, 34, 35	syl2an2r 685	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
37	26, 36	eqtr3id 2785	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
38	24, 37	eqtr3d 2773	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∩ 𝑋) = ∪ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4863  ∩ cint 4902   ↦ cmpt 5179  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   “ cima 5627  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492   Isom wiso 6493   [⊊] crpss 7667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-rpss 7668

This theorem is referenced by:  isf34lem7  10291