Theorem bdayval 33268

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 7594	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ V)
2		dmeq 5736	. . 3 ⊢ (𝑥 = 𝐴 → dom 𝑥 = dom 𝐴)
3		df-bday 33265	. . 3 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	2, 3	fvmptg 6743	. 2 ⊢ ((𝐴 ∈ No ∧ dom 𝐴 ∈ V) → ( bday ‘𝐴) = dom 𝐴)
5	1, 4	mpdan 686	1 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441  dom cdm 5519  ‘cfv 6324   No csur 33260   bday cbday 33262

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-bday 33265

This theorem is referenced by:  nofnbday  33272  fvnobday  33296  nodenselem5  33305  nodense  33309  nosupno  33316  nosupbday  33318  noetalem3  33332  noetalem4  33333