Theorem restfpw 22993

Description: The restriction of the set of finite subsets of 𝐴 is the set of finite subsets of 𝐵. (Contributed by Mario Carneiro, 18-Sep-2015.)

Assertion

Ref	Expression
restfpw	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → ((𝒫 𝐴 ∩ Fin) ↾t 𝐵) = (𝒫 𝐵 ∩ Fin))

Proof of Theorem restfpw

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 5366	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2	1	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝒫 𝐴 ∈ V)
3		inex1g 5309	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
4	2, 3	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → (𝒫 𝐴 ∩ Fin) ∈ V)
5		ssexg 5313	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐵 ∈ V)
6	5	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
7		restval 17368	. . . 4 ⊢ (((𝒫 𝐴 ∩ Fin) ∈ V ∧ 𝐵 ∈ V) → ((𝒫 𝐴 ∩ Fin) ↾t 𝐵) = ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝑥 ∩ 𝐵)))
8	4, 6, 7	syl2anc 583	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → ((𝒫 𝐴 ∩ Fin) ↾t 𝐵) = ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝑥 ∩ 𝐵)))
9		inss2 4221	. . . . . . 7 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵
10	9	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∩ 𝐵) ⊆ 𝐵)
11		elinel2 4188	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin)
12	11	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ∈ Fin)
13		inss1 4220	. . . . . . 7 ⊢ (𝑥 ∩ 𝐵) ⊆ 𝑥
14		ssfi 9168	. . . . . . 7 ⊢ ((𝑥 ∈ Fin ∧ (𝑥 ∩ 𝐵) ⊆ 𝑥) → (𝑥 ∩ 𝐵) ∈ Fin)
15	12, 13, 14	sylancl 585	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∩ 𝐵) ∈ Fin)
16		elfpw 9349	. . . . . 6 ⊢ ((𝑥 ∩ 𝐵) ∈ (𝒫 𝐵 ∩ Fin) ↔ ((𝑥 ∩ 𝐵) ⊆ 𝐵 ∧ (𝑥 ∩ 𝐵) ∈ Fin))
17	10, 15, 16	sylanbrc 582	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∩ 𝐵) ∈ (𝒫 𝐵 ∩ Fin))
18	17	fmpttd 7106	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝑥 ∩ 𝐵)):(𝒫 𝐴 ∩ Fin)⟶(𝒫 𝐵 ∩ Fin))
19	18	frnd 6715	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝑥 ∩ 𝐵)) ⊆ (𝒫 𝐵 ∩ Fin))
20	8, 19	eqsstrd 4012	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → ((𝒫 𝐴 ∩ Fin) ↾t 𝐵) ⊆ (𝒫 𝐵 ∩ Fin))
21		elfpw 9349	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ Fin))
22	21	simplbi 497	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) → 𝑥 ⊆ 𝐵)
23	22	adantl 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑥 ⊆ 𝐵)
24		df-ss 3957	. . . 4 ⊢ (𝑥 ⊆ 𝐵 ↔ (𝑥 ∩ 𝐵) = 𝑥)
25	23, 24	sylib 217	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑥 ∩ 𝐵) = 𝑥)
26	4	adantr 480	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) → (𝒫 𝐴 ∩ Fin) ∈ V)
27	6	adantr 480	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) → 𝐵 ∈ V)
28		simplr 766	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) → 𝐵 ⊆ 𝐴)
29	23, 28	sstrd 3984	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑥 ⊆ 𝐴)
30		elinel2 4188	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝐵 ∩ Fin) → 𝑥 ∈ Fin)
31	30	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑥 ∈ Fin)
32		elfpw 9349	. . . . 5 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ∈ Fin))
33	29, 31, 32	sylanbrc 582	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑥 ∈ (𝒫 𝐴 ∩ Fin))
34		elrestr 17370	. . . 4 ⊢ (((𝒫 𝐴 ∩ Fin) ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∩ 𝐵) ∈ ((𝒫 𝐴 ∩ Fin) ↾t 𝐵))
35	26, 27, 33, 34	syl3anc 1368	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) → (𝑥 ∩ 𝐵) ∈ ((𝒫 𝐴 ∩ Fin) ↾t 𝐵))
36	25, 35	eqeltrrd 2826	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 ∈ (𝒫 𝐵 ∩ Fin)) → 𝑥 ∈ ((𝒫 𝐴 ∩ Fin) ↾t 𝐵))
37	20, 36	eqelssd 3995	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴) → ((𝒫 𝐴 ∩ Fin) ↾t 𝐵) = (𝒫 𝐵 ∩ Fin))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∩ cin 3939   ⊆ wss 3940  𝒫 cpw 4594   ↦ cmpt 5221  ran crn 5667  (class class class)co 7401  Fincfn 8934   ↾_t crest 17362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1o 8461  df-en 8935  df-fin 8938  df-rest 17364

This theorem is referenced by: (None)