Theorem bdayval 27987

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 7902	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ V)
2		dmeq 5885	. . 3 ⊢ (𝑥 = 𝐴 → dom 𝑥 = dom 𝐴)
3		df-bday 27984	. . 3 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	2, 3	fvmptg 6983	. 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 2145  Vcvv 3451  dom cdm 5651  ‘cfv 6531   No csur 27979   bday cbday 27981

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fv 6539  df-bday 27984

This theorem is used by:  nofnbday  27991  fvnobday  28017  nodenselem5  28027  nodense  28031  nosupno  28042  nosupbday  28044  noinfno  28057  noinfbday  28059  noetasuplem4  28075  noetainflem4  28079  onnobdayg  44389  bdaybndex  44390  bdaybndbday  44391