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Theorem ressid2 17413

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

**Hypotheses**
Ref	Expression
ressbas.r	⊢ 𝑅 = (𝑊 ↾_s 𝐴)
ressbas.b	⊢ 𝐵 = (Base‘𝑊)

**Assertion**
Ref	Expression
ressid2	⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊)

**Proof of Theorem ressid2**
Step	Hyp	Ref	Expression
1		ressbas.r	. . . 4 ⊢ 𝑅 = (𝑊 ↾_s 𝐴)
2		ressbas.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
3	1, 2	ressval 17412	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
4		iftrue 4487	. . 3 ⊢ (𝐵 ⊆ 𝐴 → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)) = 𝑊)
5	3, 4	sylan9eqr 2817	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌)) → 𝑅 = 𝑊)
6	5	3impb 1132	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ⊆ wss 3895 ∩ cin 4089 ifcif 4481 ⟨cop 4589 ‘cfv 6538 (class class class)co 7420 sSet csts 17342 ndxcnx 17372 Basecbs 17388 ↾_s cress 17409

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-iota 6494 df-fun 6540 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-ress 17410

This theorem is used by: ressbas 17415 resseqnbas 17421 ress0 17422 ressid 17423 ressinbas 17424 ressress 17426 rescabs 18009 0symgefmndeq 19609 snsymgefmndeq 19610 mgpress 20371 psgnghm2 21888 evl1maprhm 22698 resvsca 33936 extdg1id 34341

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