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Theorem elno2 32682
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 32677 . . 3 (𝐴 No → Fun 𝐴)
2 nodmon 32678 . . 3 (𝐴 No → dom 𝐴 ∈ On)
3 norn 32679 . . 3 (𝐴 No → ran 𝐴 ⊆ {1o, 2o})
41, 2, 33jca 1108 . 2 (𝐴 No → (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
5 simp2 1117 . . . 4 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → dom 𝐴 ∈ On)
6 simpl 475 . . . . . . . 8 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → Fun 𝐴)
76funfnd 6213 . . . . . . 7 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → 𝐴 Fn dom 𝐴)
87anim1i 605 . . . . . 6 (((Fun 𝐴 ∧ dom 𝐴 ∈ On) ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
983impa 1090 . . . . 5 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
10 df-f 6186 . . . . 5 (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
119, 10sylibr 226 . . . 4 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴:dom 𝐴⟶{1o, 2o})
12 feq2 6320 . . . . 5 (𝑥 = dom 𝐴 → (𝐴:𝑥⟶{1o, 2o} ↔ 𝐴:dom 𝐴⟶{1o, 2o}))
1312rspcev 3529 . . . 4 ((dom 𝐴 ∈ On ∧ 𝐴:dom 𝐴⟶{1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
145, 11, 13syl2anc 576 . . 3 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
15 elno 32674 . . 3 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
1614, 15sylibr 226 . 2 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴 No )
174, 16impbii 201 1 (𝐴 No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  w3a 1068  wcel 2050  wrex 3083  wss 3823  {cpr 4437  dom cdm 5401  ran crn 5402  Oncon0 6023  Fun wfun 6176   Fn wfn 6177  wf 6178  1oc1o 7892  2oc2o 7893   No csur 32668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-no 32671
This theorem is referenced by:  elno3  32683  noextend  32694  noextendseq  32695  nosupno  32724
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