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Theorem elno3 31933
Description: Another condition for membership in No . (Contributed by Scott Fenton, 14-Apr-2012.)
Assertion
Ref Expression
elno3 (𝐴 No ↔ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ∧ dom 𝐴 ∈ On))

Proof of Theorem elno3
StepHypRef Expression
1 3anan32 1068 . 2 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) ∧ dom 𝐴 ∈ On))
2 elno2 31932 . 2 (𝐴 No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
3 df-f 5930 . . . 4 (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
4 funfn 5956 . . . . 5 (Fun 𝐴𝐴 Fn dom 𝐴)
54anbi1i 731 . . . 4 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
63, 5bitr4i 267 . . 3 (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ↔ (Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
76anbi1i 731 . 2 ((𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ∧ dom 𝐴 ∈ On) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) ∧ dom 𝐴 ∈ On))
81, 2, 73bitr4i 292 1 (𝐴 No ↔ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ∧ dom 𝐴 ∈ On))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1054  wcel 2030  wss 3607  {cpr 4212  dom cdm 5143  ran crn 5144  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  wf 5922  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-no 31921
This theorem is referenced by:  noxp1o  31941  noseponlem  31942
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