Theorem bdayval 27626

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 7845	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ V)
2		dmeq 5852	. . 3 ⊢ (𝑥 = 𝐴 → dom 𝑥 = dom 𝐴)
3		df-bday 27622	. . 3 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	2, 3	fvmptg 6939	. 2 ⊢ ((𝐴 ∈ No ∧ dom 𝐴 ∈ V) → ( bday ‘𝐴) = dom 𝐴)
5	1, 4	mpdan 688	1 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  dom cdm 5624  ‘cfv 6492   No csur 27617   bday cbday 27619

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 5231  ax-pr 5370  ax-un 7682

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-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-bday 27622

This theorem is referenced by:  nofnbday  27630  fvnobday  27656  nodenselem5  27666  nodense  27670  nosupno  27681  nosupbday  27683  noinfno  27696  noinfbday  27698  noetasuplem4  27714  noetainflem4  27718  onnobdayg  43875  bdaybndex  43876  bdaybndbday  43877