ressval2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ressval2		Structured version Visualization version GIF version

Theorem ressval2 17425

Description: Value of nontrivial structure 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
ressval2	⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))

**Proof of Theorem ressval2**
Step	Hyp	Ref	Expression
1		ressbas.r	. . . 4 ⊢ 𝑅 = (𝑊 ↾_s 𝐴)
2		ressbas.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
3	1, 2	ressval 17423	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
4		iffalse 4493	. . 3 ⊢ (¬ 𝐵 ⊆ 𝐴 → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
5	3, 4	sylan9eqr 2820	. 2 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌)) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
6	5	3impb 1132	1 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ⊆ wss 3899 ∩ cin 4091 ifcif 4484 ⟨cop 4837 ‘cfv 6546 (class class class)co 7427 sSet csts 17353 ndxcnx 17383 Basecbs 17399 ↾_s cress 17420

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 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pr 5394

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 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3416 df-v 3455 df-sbc 3742 df-ss 3900 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-br 5106 df-opab 5170 df-id 5549 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-iota 6500 df-fun 6548 df-fv 6554 df-ov 7430 df-oprab 7431 df-mpo 7432 df-ress 17421

This theorem is used by: ressbas 17426 resseqnbas 17432 ressinbas 17435 ressval3d 17436 ressress 17437 rescabs 18020 symgvalstruct 19623 mgpress 20382 resssra 34272

W3C validator