Theorem stdbdmetval 24700

Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypothesis

Ref	Expression
stdbdmet.1	⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))

Assertion

Ref	Expression
stdbdmetval	⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐷(𝑥, 𝑦)   𝑉(𝑥, 𝑦)

Proof of Theorem stdbdmetval

Step	Hyp	Ref	Expression
1		ovex 7449	. . . 4 ⊢ (𝐴𝐶𝐵) ∈ V
2		ifexg 4539	. . . 4 ⊢ (((𝐴𝐶𝐵) ∈ V ∧ 𝑅 ∈ 𝑉) → if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V)
3	1, 2	mpan 703	. . 3 ⊢ (𝑅 ∈ 𝑉 → if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V)
4		oveq12 7425	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐶𝑦) = (𝐴𝐶𝐵))
5	4	breq1d 5121	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥𝐶𝑦) ≤ 𝑅 ↔ (𝐴𝐶𝐵) ≤ 𝑅))
6	5, 4	ifbieq1d 4514	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
7		stdbdmet.1	. . . 4 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if((𝑥𝐶𝑦) ≤ 𝑅, (𝑥𝐶𝑦), 𝑅))
8	6, 7	ovmpoga 7570	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
9	3, 8	syl3an3 1183	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑅 ∈ 𝑉) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
10	9	3comr 1143	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ifcif 4489   class class class wbr 5111  (class class class)co 7416   ∈ cmpo 7418   ≤ cle 11255

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421

This theorem is used by:  stdbdbl  24703