Theorem ressval 17404

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

Proof of Theorem ressval

Dummy variables 𝑤 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ressbas.r	. 2 ⊢ 𝑅 = (𝑊 ↾s 𝐴)
2		elex 3472	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		elex 3472	. . 3 ⊢ (𝐴 ∈ 𝑌 → 𝐴 ∈ V)
4		simpl 488	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → 𝑊 ∈ V)
5		ovex 7451	. . . . 5 ⊢ (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) ∈ V
6		ifcl 4528	. . . . 5 ⊢ ((𝑊 ∈ V ∧ (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) ∈ V) → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)) ∈ V)
7	4, 5, 6	sylancl 598	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)) ∈ V)
8		simpl 488	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
9	8	fveq2d 6887	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
10		ressbas.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊)
11	9, 10	eqtr4di 2814	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
12		simpr 490	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
13	11, 12	sseq12d 3964	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → ((Base‘𝑤) ⊆ 𝑎 ↔ 𝐵 ⊆ 𝐴))
14	12, 11	ineq12d 4167	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (𝑎 ∩ (Base‘𝑤)) = (𝐴 ∩ 𝐵))
15	14	opeq2d 4840	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)
16	8, 15	oveq12d 7436	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
17	13, 8, 16	ifbieq12d 4511	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
18		df-ress 17402	. . . . 5 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
19	17, 18	ovmpoga 7572	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V ∧ if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)) ∈ V) → (𝑊 ↾s 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
20	7, 19	mpd3an3 1491	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
21	2, 3, 20	syl2an 608	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → (𝑊 ↾s 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
22	1, 21	eqtrid 2808	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  ifcif 4482  ⟨cop 4590  ‘cfv 6537  (class class class)co 7418   sSet csts 17334  ndxcnx 17364  Basecbs 17380   ↾_s cress 17401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-ress 17402

This theorem is used by:  ressid2  17405  ressval2  17406  wunress  17420