Theorem elno3 31808

Description: Another condition for membership in No. (Contributed by Scott Fenton, 14-Apr-2012.)

Assertion

Ref	Expression
elno3	|- ( A e. No <-> ( A : dom A --> { 1o , 2o } /\ dom A e. On ) )

Proof of Theorem elno3

Step	Hyp	Ref	Expression
1		3anan32 1050	. 2 |- ( ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) <-> ( ( Fun A /\ ran A C_ { 1o , 2o } ) /\ dom A e. On ) )
2		elno2 31807	. 2 |- ( A e. No <-> ( Fun A /\ dom A e. On /\ ran A C_ { 1o , 2o } ) )
3		df-f 5892	. . . 4 |- ( A : dom A --> { 1o , 2o } <-> ( A Fn dom A /\ ran A C_ { 1o , 2o } ) )
4		funfn 5918	. . . . 5 |- ( Fun A <-> A Fn dom A )
5	4	anbi1i 731	. . . 4 |- ( ( Fun A /\ ran A C_ { 1o , 2o } ) <-> ( A Fn dom A /\ ran A C_ { 1o , 2o } ) )
6	3, 5	bitr4i 267	. . 3 |- ( A : dom A --> { 1o , 2o } <-> ( Fun A /\ ran A C_ { 1o , 2o } ) )
7	6	anbi1i 731	. 2 |- ( ( A : dom A --> { 1o , 2o } /\ dom A e. On ) <-> ( ( Fun A /\ ran A C_ { 1o , 2o } ) /\ dom A e. On ) )
8	1, 2, 7	3bitr4i 292	1 |- ( A e. No <-> ( A : dom A --> { 1o , 2o } /\ dom A e. On ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   C_ wss 3574   {cpr 4179   dom cdm 5114   ran crn 5115   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -->wf 5884   1oc1o 7553   2oc2o 7554   Nocsur 31793

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-no 31796

This theorem is referenced by:  noxp1o  31816  noseponlem  31817