ressbasid - Intuitionistic Logic Explorer

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Theorem ressbasid 13424

Description: The trivial structure restriction leaves the base set unchanged. (Contributed by Jim Kingdon, 29-Apr-2025.)

**Hypothesis**
Ref	Expression
ressbasid.b	⊢ 𝐵 = (Base‘𝑊)

**Assertion**
Ref	Expression
ressbasid	⊢ (𝑊 ∈ 𝑉 → (Base‘(𝑊 ↾_s 𝐵)) = 𝐵)

**Proof of Theorem ressbasid**
Step	Hyp	Ref	Expression
1		eqidd 2239	. . 3 ⊢ (𝑊 ∈ 𝑉 → (𝑊 ↾_s 𝐵) = (𝑊 ↾_s 𝐵))
2		ressbasid.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
3	2	a1i 9	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝐵 = (Base‘𝑊))
4		id 19	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ 𝑉)
5		basfn 13411	. . . . 5 ⊢ Base Fn V
6		elex 2833	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
7		funfvex 5712	. . . . . 6 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
8	7	funfni 5483	. . . . 5 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
9	5, 6, 8	sylancr 418	. . . 4 ⊢ (𝑊 ∈ 𝑉 → (Base‘𝑊) ∈ V)
10	2, 9	eqeltrid 2325	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝐵 ∈ V)
11	1, 3, 4, 10	ressbasd 13421	. 2 ⊢ (𝑊 ∈ 𝑉 → (𝐵 ∩ 𝐵) = (Base‘(𝑊 ↾_s 𝐵)))
12		inidm 3440	. 2 ⊢ (𝐵 ∩ 𝐵) = 𝐵
13	11, 12	eqtr3di 2286	1 ⊢ (𝑊 ∈ 𝑉 → (Base‘(𝑊 ↾_s 𝐵)) = 𝐵)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 = wceq 1402 ∈ wcel 2209 Vcvv 2821 ∩ cin 3219 Fn wfn 5372 ‘cfv 5377 (class class class)co 6085 Basecbs 13352 ↾_s cress 13353

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1re 8273 ax-addrcl 8276

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-iota 5337 df-fun 5379 df-fn 5380 df-fv 5385 df-ov 6088 df-oprab 6089 df-mpo 6090 df-inn 9305 df-ndx 13355 df-slot 13356 df-base 13358 df-sets 13359 df-iress 13360

This theorem is used by: rlmscabas 14797

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