Theorem stdbdmetval 24470

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 7401	. . . 4 ⊢ (𝐴𝐶𝐵) ∈ V
2		ifexg 4531	. . . 4 ⊢ (((𝐴𝐶𝐵) ∈ V ∧ 𝑅 ∈ 𝑉) → if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V)
3	1, 2	mpan 691	. . 3 ⊢ (𝑅 ∈ 𝑉 → if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V)
4		oveq12 7377	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐶𝑦) = (𝐴𝐶𝐵))
5	4	breq1d 5110	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥𝐶𝑦) ≤ 𝑅 ↔ (𝐴𝐶𝐵) ≤ 𝑅))
6	5, 4	ifbieq1d 4506	. . . 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 7522	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅) ∈ V) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
9	3, 8	syl3an3 1166	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑅 ∈ 𝑉) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))
10	9	3comr 1126	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = if((𝐴𝐶𝐵) ≤ 𝑅, (𝐴𝐶𝐵), 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ifcif 4481   class class class wbr 5100  (class class class)co 7368   ∈ cmpo 7370   ≤ cle 11179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373

This theorem is referenced by:  stdbdbl  24473