Theorem bdayval 27576

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 7841	. 2 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ V)
2		dmeq 5850	. . 3 ⊢ (𝑥 = 𝐴 → dom 𝑥 = dom 𝐴)
3		df-bday 27572	. . 3 ⊢ bday = (𝑥 ∈ No ↦ dom 𝑥)
4	2, 3	fvmptg 6932	. 2 ⊢ ((𝐴 ∈ No ∧ dom 𝐴 ∈ V) → ( bday ‘𝐴) = dom 𝐴)
5	1, 4	mpdan 687	1 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3438  dom cdm 5623  ‘cfv 6486   No csur 27567   bday cbday 27569

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fv 6494  df-bday 27572

This theorem is referenced by:  nofnbday  27580  fvnobday  27606  nodenselem5  27616  nodense  27620  nosupno  27631  nosupbday  27633  noinfno  27646  noinfbday  27648  noetasuplem4  27664  noetainflem4  27668  onnobdayg  43403  bdaybndex  43404  bdaybndbday  43405