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Theorem ressval2 17181

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 17179	. . 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 2786	. 2 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌)) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
6	5	3impb 1114	1 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∩ cin 3910 ⊆ wss 3911 ifcif 4484 ⟨cop 4591 ‘cfv 6499 (class class class)co 7369 sSet csts 17109 ndxcnx 17139 Basecbs 17155 ↾_s cress 17176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pr 5382

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6452 df-fun 6501 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-ress 17177

This theorem is referenced by: ressbas 17182 resseqnbas 17188 ressinbas 17191 ressval3d 17192 ressress 17193 rescabs 17775 symgvalstruct 19311 mgpress 20070 resssra 33576

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