residfi - Metamath Proof Explorer

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Theorem residfi 9302

Description: A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)

**Assertion**
Ref	Expression
residfi	⊢ (( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)

**Proof of Theorem residfi**
Step	Hyp	Ref	Expression
1		dmresi 6056	. . 3 ⊢ dom ( I ↾ 𝐴) = 𝐴
2		dmfi 9299	. . 3 ⊢ (( I ↾ 𝐴) ∈ Fin → dom ( I ↾ 𝐴) ∈ Fin)
3	1, 2	eqeltrrid 2870	. 2 ⊢ (( I ↾ 𝐴) ∈ Fin → 𝐴 ∈ Fin)
4		funi 6572	. . . 4 ⊢ Fun I
5		funfn 6570	. . . 4 ⊢ (Fun I ↔ I Fn dom I )
6	4, 5	mpbi 233	. . 3 ⊢ I Fn dom I
7		resfnfinfin 9301	. . 3 ⊢ (( I Fn dom I ∧ 𝐴 ∈ Fin) → ( I ↾ 𝐴) ∈ Fin)
8	6, 7	mpan 703	. 2 ⊢ (𝐴 ∈ Fin → ( I ↾ 𝐴) ∈ Fin)
9	3, 8	impbii 212	1 ⊢ (( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∈ wcel 2146 I cid 5557 dom cdm 5663 ↾ cres 5665 Fun wfun 6534 Fn wfn 6535 Fincfn 8949

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7869 df-1st 7992 df-2nd 7993 df-1o 8459 df-en 8950 df-dom 8951 df-fin 8953

This theorem is used by: fusgrfisstep 29737