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Theorem ressid 12479

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 3167	. 2 ⊢ 𝐵 ⊆ 𝐵
2		id 19	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ 𝑋)
3		ressid.1	. . 3 ⊢ 𝐵 = (Base‘𝑊)
4		baseid 12469	. . . . 5 ⊢ Base = Slot (Base‘ndx)
5		basendxnn 12471	. . . . . 6 ⊢ (Base‘ndx) ∈ ℕ
6	5	a1i 9	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → (Base‘ndx) ∈ ℕ)
7	4, 2, 6	strnfvnd 12436	. . . 4 ⊢ (𝑊 ∈ 𝑋 → (Base‘𝑊) = (𝑊‘(Base‘ndx)))
8		fvexg 5515	. . . . 5 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘ndx) ∈ ℕ) → (𝑊‘(Base‘ndx)) ∈ V)
9	5, 8	mpan2 423	. . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝑊‘(Base‘ndx)) ∈ V)
10	7, 9	eqeltrd 2247	. . 3 ⊢ (𝑊 ∈ 𝑋 → (Base‘𝑊) ∈ V)
11	3, 10	eqeltrid 2257	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝐵 ∈ V)
12		eqid 2170	. . 3 ⊢ (𝑊 ↾_s 𝐵) = (𝑊 ↾_s 𝐵)
13	12, 3	ressid2 12477	. 2 ⊢ ((𝐵 ⊆ 𝐵 ∧ 𝑊 ∈ 𝑋 ∧ 𝐵 ∈ V) → (𝑊 ↾_s 𝐵) = 𝑊)
14	1, 2, 11, 13	mp3an2i 1337	1 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾_s 𝐵) = 𝑊)

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 Vcvv 2730 ⊆ wss 3121 ‘cfv 5198 (class class class)co 5853 ℕcn 8878 ndxcnx 12413 Basecbs 12416 ↾_s cress 12417

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-inn 8879 df-ndx 12419 df-slot 12420 df-base 12422 df-ress 12424

This theorem is referenced by: (None)

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