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Theorem residfi 9241

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 6011	. . 3 ⊢ dom ( I ↾ 𝐴) = 𝐴
2		dmfi 9238	. . 3 ⊢ (( I ↾ 𝐴) ∈ Fin → dom ( I ↾ 𝐴) ∈ Fin)
3	1, 2	eqeltrrid 2842	. 2 ⊢ (( I ↾ 𝐴) ∈ Fin → 𝐴 ∈ Fin)
4		funi 6524	. . . 4 ⊢ Fun I
5		funfn 6522	. . . 4 ⊢ (Fun I ↔ I Fn dom I )
6	4, 5	mpbi 230	. . 3 ⊢ I Fn dom I
7		resfnfinfin 9240	. . 3 ⊢ (( I Fn dom I ∧ 𝐴 ∈ Fin) → ( I ↾ 𝐴) ∈ Fin)
8	6, 7	mpan 691	. 2 ⊢ (𝐴 ∈ Fin → ( I ↾ 𝐴) ∈ Fin)
9	3, 8	impbii 209	1 ⊢ (( I ↾ 𝐴) ∈ Fin ↔ 𝐴 ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∈ wcel 2114 I cid 5518 dom cdm 5624 ↾ cres 5626 Fun wfun 6486 Fn wfn 6487 Fincfn 8886

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 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-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 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-om 7811 df-1st 7935 df-2nd 7936 df-1o 8398 df-en 8887 df-dom 8888 df-fin 8890

This theorem is referenced by: fusgrfisstep 29412