Theorem bdayval 31801

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

Assertion

Ref	Expression
bdayval	|- ( A e. No -> ( bday ` A ) = dom A )

Proof of Theorem bdayval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmexg 7097	. 2 |- ( A e. No -> dom A e. _V )
2		dmeq 5324	. . 3 |- ( x = A -> dom x = dom A )
3		df-bday 31798	. . 3 |- bday = ( x e. No |-> dom x )
4	2, 3	fvmptg 6280	. 2 |- ( ( A e. No /\ dom A e. _V ) -> ( bday ` A ) = dom A )
5	1, 4	mpdan 702	1 |- ( A e. No -> ( bday ` A ) = dom A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   dom cdm 5114   ` cfv 5888   Nocsur 31793   bdaycbday 31795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-bday 31798

This theorem is referenced by:  nofnbday  31805  fvnobday  31829  nodenselem5  31838  nodense  31842  nosupno  31849  nosupbday  31851  noetalem3  31865  noetalem4  31866