resiexd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > resiexd		Structured version Visualization version GIF version

Theorem resiexd 7253

Description: The restriction of the identity relation to a set is a set. (Contributed by AV, 15-Feb-2020.)

**Hypothesis**
Ref	Expression
resiexd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)

**Assertion**
Ref	Expression
resiexd	⊢ (𝜑 → ( I ↾ 𝐵) ∈ V)

**Proof of Theorem resiexd**
Step	Hyp	Ref	Expression
1		funi 6610	. 2 ⊢ Fun I
2		resiexd.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
3		resfunexg 7252	. 2 ⊢ ((Fun I ∧ 𝐵 ∈ 𝑉) → ( I ↾ 𝐵) ∈ V)
4	1, 2, 3	sylancr 586	1 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3488 I cid 5592 ↾ cres 5702 Fun wfun 6567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581