Theorem ressval 17260

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 3474	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		elex 3474	. . 3 ⊢ (𝐴 ∈ 𝑌 → 𝐴 ∈ V)
4		simpl 486	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → 𝑊 ∈ V)
5		ovex 7424	. . . . 5 ⊢ (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) ∈ V
6		ifcl 4523	. . . . 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 595	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)) ∈ V)
8		simpl 486	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
9	8	fveq2d 6866	. . . . . . . 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 488	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
13	11, 12	sseq12d 3967	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → ((Base‘𝑤) ⊆ 𝑎 ↔ 𝐵 ⊆ 𝐴))
14	12, 11	ineq12d 4171	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (𝑎 ∩ (Base‘𝑤)) = (𝐴 ∩ 𝐵))
15	14	opeq2d 4835	. . . . . . 7 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩ = ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)
16	8, 15	oveq12d 7409	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑎 = 𝐴) → (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
17	13, 8, 16	ifbieq12d 4506	. . . . 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 17258	. . . . 5 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
19	17, 18	ovmpoga 7545	. . . 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 1482	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩)))
21	2, 3, 20	syl2an 605	. 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)⟩)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ifcif 4477  ⟨cop 4585  ‘cfv 6516  (class class class)co 7391   sSet csts 17190  ndxcnx 17220  Basecbs 17236   ↾_s cress 17257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-ress 17258

This theorem is referenced by:  ressid2  17261  ressval2  17262  wunress  17276