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Theorem residfi 9238

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 6004	. . 3 ⊢ dom ( I ↾ 𝐴) = 𝐴
2		dmfi 9235	. . 3 ⊢ (( I ↾ 𝐴) ∈ Fin → dom ( I ↾ 𝐴) ∈ Fin)
3	1, 2	eqeltrrid 2844	. 2 ⊢ (( I ↾ 𝐴) ∈ Fin → 𝐴 ∈ Fin)
4		funi 6517	. . . 4 ⊢ Fun I
5		funfn 6515	. . . 4 ⊢ (Fun I ↔ I Fn dom I )
6	4, 5	mpbi 231	. . 3 ⊢ I Fn dom I
7		resfnfinfin 9237	. . 3 ⊢ (( I Fn dom I ∧ 𝐴 ∈ Fin) → ( I ↾ 𝐴) ∈ Fin)
8	6, 7	mpan 696	. 2 ⊢ (𝐴 ∈ Fin → ( I ↾ 𝐴) ∈ Fin)
9	3, 8	impbii 210	1 ⊢ (( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 207 ∈ wcel 2119 I cid 5512 dom cdm 5618 ↾ cres 5620 Fun wfun 6479 Fn wfn 6480 Fincfn 8883

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-om 7807 df-1st 7931 df-2nd 7932 df-1o 8395 df-en 8884 df-dom 8885 df-fin 8887

This theorem is referenced by: fusgrfisstep 29416