Theorem isf34lem5 9653

Description: Lemma for isfin3-4 9657. (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 5824	. . . . . . 7 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		compss.a	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥))
3	2	isf34lem2 9648	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
4	3	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴)
5	4	frnd 6396	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ran 𝐹 ⊆ 𝒫 𝐴)
6	1, 5	sstrid 3906	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ 𝑋) ⊆ 𝒫 𝐴)
7		simprl 767	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ 𝒫 𝐴)
8	4	fdmd 6398	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → dom 𝐹 = 𝒫 𝐴)
9	7, 8	sseqtr4d 3935	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ dom 𝐹)
10		sseqin2 4118	. . . . . . . . 9 ⊢ (𝑋 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑋) = 𝑋)
11	9, 10	sylib 219	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (dom 𝐹 ∩ 𝑋) = 𝑋)
12		simprr 769	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ≠ ∅)
13	11, 12	eqnetrd 3053	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (dom 𝐹 ∩ 𝑋) ≠ ∅)
14		imadisj 5831	. . . . . . . 8 ⊢ ((𝐹 “ 𝑋) = ∅ ↔ (dom 𝐹 ∩ 𝑋) = ∅)
15	14	necon3bii 3038	. . . . . . 7 ⊢ ((𝐹 “ 𝑋) ≠ ∅ ↔ (dom 𝐹 ∩ 𝑋) ≠ ∅)
16	13, 15	sylibr 235	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ 𝑋) ≠ ∅)
17	6, 16	jca 512	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ∧ (𝐹 “ 𝑋) ≠ ∅))
18	2	isf34lem4 9652	. . . . 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 9650	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
21	20	adantrr 713	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋)
22	21	inteqd 4793	. . . 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 6549	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = (𝐹‘∩ 𝑋))
25	2	compsscnv 9646	. . . 4 ⊢ ◡𝐹 = 𝐹
26	25	fveq1i 6546	. . 3 ⊢ (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋)))
27	2	compssiso 9649	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴))
28		isof1o 6946	. . . . 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 4927	. . . . . 6 ⊢ ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴)
31	6, 30	sylib 219	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∪ (𝐹 “ 𝑋) ⊆ 𝐴)
32		elpw2g 5145	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴))
33	32	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴))
34	31, 33	mpbird 258	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴)
35		f1ocnvfv1 6905	. . . 4 ⊢ ((𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴 ∧ ∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴) → (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
36	29, 34, 35	syl2an2r 681	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
37	26, 36	syl5eqr 2847	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋))
38	24, 37	eqtr3d 2835	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∩ 𝑋) = ∪ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083   ≠ wne 2986   ∖ cdif 3862   ∩ cin 3864   ⊆ wss 3865  ∅c0 4217  𝒫 cpw 4459  ∪ cuni 4751  ∩ cint 4788   ↦ cmpt 5047  ^◡ccnv 5449  dom cdm 5450  ran crn 5451   “ cima 5453  ⟶wf 6228  –1-1-onto→wf1o 6231  ‘cfv 6232   Isom wiso 6233   [⊊] crpss 7313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-int 4789  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-rpss 7314

This theorem is referenced by:  isf34lem7  9654