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Theorem ressid 12190

Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)

**Hypothesis**
Ref	Expression
ressid.1	⊢ 𝐵 = (Base‘𝑊)

**Assertion**
Ref	Expression
ressid	⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾_s 𝐵) = 𝑊)

**Proof of Theorem ressid**
Step	Hyp	Ref	Expression
1		ssid 3144	. 2 ⊢ 𝐵 ⊆ 𝐵
2		id 19	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ 𝑋)
3		ressid.1	. . 3 ⊢ 𝐵 = (Base‘𝑊)
4		baseid 12182	. . . . 5 ⊢ Base = Slot (Base‘ndx)
5		basendxnn 12184	. . . . . 6 ⊢ (Base‘ndx) ∈ ℕ
6	5	a1i 9	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → (Base‘ndx) ∈ ℕ)
7	4, 2, 6	strnfvnd 12149	. . . 4 ⊢ (𝑊 ∈ 𝑋 → (Base‘𝑊) = (𝑊‘(Base‘ndx)))
8		fvexg 5480	. . . . 5 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘ndx) ∈ ℕ) → (𝑊‘(Base‘ndx)) ∈ V)
9	5, 8	mpan2 422	. . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝑊‘(Base‘ndx)) ∈ V)
10	7, 9	eqeltrd 2231	. . 3 ⊢ (𝑊 ∈ 𝑋 → (Base‘𝑊) ∈ V)
11	3, 10	eqeltrid 2241	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝐵 ∈ V)
12		eqid 2154	. . 3 ⊢ (𝑊 ↾_s 𝐵) = (𝑊 ↾_s 𝐵)
13	12, 3	ressid2 12188	. 2 ⊢ ((𝐵 ⊆ 𝐵 ∧ 𝑊 ∈ 𝑋 ∧ 𝐵 ∈ V) → (𝑊 ↾_s 𝐵) = 𝑊)
14	1, 2, 11, 13	mp3an2i 1321	1 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾_s 𝐵) = 𝑊)

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1332 ∈ wcel 2125 Vcvv 2709 ⊆ wss 3098 ‘cfv 5163 (class class class)co 5814 ℕcn 8812 ndxcnx 12126 Basecbs 12129 ↾_s cress 12130

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-cnex 7802 ax-resscn 7803 ax-1re 7805 ax-addrcl 7808

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-ral 2437 df-rex 2438 df-rab 2441 df-v 2711 df-sbc 2934 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-br 3962 df-opab 4022 df-mpt 4023 df-id 4248 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-iota 5128 df-fun 5165 df-fv 5171 df-ov 5817 df-oprab 5818 df-mpo 5819 df-inn 8813 df-ndx 12132 df-slot 12133 df-base 12135 df-ress 12137

This theorem is referenced by: (None)

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