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Theorem elno2 27714
Description: An alternative condition for membership in No . (Contributed by Scott Fenton, 21-Mar-2012.)
Assertion
Ref Expression
elno2 (𝐴 No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))

Proof of Theorem elno2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nofun 27709 . . 3 (𝐴 No → Fun 𝐴)
2 nodmon 27710 . . 3 (𝐴 No → dom 𝐴 ∈ On)
3 norn 27711 . . 3 (𝐴 No → ran 𝐴 ⊆ {1o, 2o})
41, 2, 33jca 1127 . 2 (𝐴 No → (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
5 simp2 1136 . . . 4 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → dom 𝐴 ∈ On)
6 simpl 482 . . . . . . . 8 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → Fun 𝐴)
76funfnd 6599 . . . . . . 7 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → 𝐴 Fn dom 𝐴)
87anim1i 615 . . . . . 6 (((Fun 𝐴 ∧ dom 𝐴 ∈ On) ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
983impa 1109 . . . . 5 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
10 df-f 6567 . . . . 5 (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
119, 10sylibr 234 . . . 4 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴:dom 𝐴⟶{1o, 2o})
12 feq2 6718 . . . . 5 (𝑥 = dom 𝐴 → (𝐴:𝑥⟶{1o, 2o} ↔ 𝐴:dom 𝐴⟶{1o, 2o}))
1312rspcev 3622 . . . 4 ((dom 𝐴 ∈ On ∧ 𝐴:dom 𝐴⟶{1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
145, 11, 13syl2anc 584 . . 3 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
15 elno 27705 . . 3 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
1614, 15sylibr 234 . 2 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴 No )
174, 16impbii 209 1 (𝐴 No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086  wcel 2106  wrex 3068  wss 3963  {cpr 4633  dom cdm 5689  ran crn 5690  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  wf 6559  1oc1o 8498  2oc2o 8499   No csur 27699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-no 27702
This theorem is referenced by:  elno3  27715  noextend  27726  noextendseq  27727  nosupno  27763  noinfno  27778  onnog  43419
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