Theorem bdayval 28035

Description: The value of the birthday function within the surreals. (Contributed by Scott Fenton, 14-Jun-2011.)

Assertion

Ref	Expression
bdayval	⊢ (𝐴 ∈ No → (bday‘𝐴) = dom 𝐴)

Proof of Theorem bdayval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmexg 7937	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ V)
2		dmeq 5892	. . 3 ⊢ (𝑥 = 𝐴 → dom 𝑥 = dom 𝐴)
3		df-bday 28032	. . 3 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	2, 3	fvmptg 7014	. 2 ⊢ ((𝐴 ∈ No ∧ dom 𝐴 ∈ V) → (bday‘𝐴) = dom 𝐴)
5	1, 4	mpdan 700	1 ⊢ (𝐴 ∈ No → (bday‘𝐴) = dom 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  dom cdm 5655  ‘cfv 6558  Nocsur 28027  bdaycbday 28029

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fv 6566  df-bday 28032

This theorem is used by:  nofnbday  28039  fvnobday  28065  nodenselem5  28075  nodense  28079  nosupno  28090  nosupbday  28092  noinfno  28105  noinfbday  28107  noetasuplem4  28123  noetainflem4  28127  onnobdayg  44623  bdaybndex  44624  bdaybndbday  44625