Theorem metdsval 24787

Description: Value of the "distance to a set" function. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.) (Revised by AV, 30-Sep-2020.)

Hypothesis

Ref	Expression
metdscn.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))

Assertion

Ref	Expression
metdsval	⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐷,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem metdsval

Step	Hyp	Ref	Expression
1		oveq1 7412	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦))
2	1	mpteq2dv 5215	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) = (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)))
3	2	rneqd 5918	. . 3 ⊢ (𝑥 = 𝐴 → ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) = ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)))
4	3	infeq1d 9490	. 2 ⊢ (𝑥 = 𝐴 → inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) = inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
5		metdscn.f	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))
6		xrltso 13157	. . 3 ⊢ < Or ℝ*
7	6	infex 9507	. 2 ⊢ inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ) ∈ V
8	4, 5, 7	fvmpt 6986	1 ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ↦ cmpt 5201  ran crn 5655  ‘cfv 6531  (class class class)co 7405  infcinf 9453  ℝ^*cxr 11268   < clt 11269

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-pre-lttri 11203  ax-pre-lttrn 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-sup 9454  df-inf 9455  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274

This theorem is referenced by:  metdsge  24789  lebnumlem1  24911  lebnumlem3  24913