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Theorem elno2 31932
 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 31927 . . 3 (𝐴 No → Fun 𝐴)
2 nodmon 31928 . . 3 (𝐴 No → dom 𝐴 ∈ On)
3 norn 31929 . . 3 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
41, 2, 33jca 1261 . 2 (𝐴 No → (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
5 simp2 1082 . . . 4 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → dom 𝐴 ∈ On)
6 simpl 472 . . . . . . . 8 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → Fun 𝐴)
7 eqidd 2652 . . . . . . . 8 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → dom 𝐴 = dom 𝐴)
8 df-fn 5929 . . . . . . . 8 (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴))
96, 7, 8sylanbrc 699 . . . . . . 7 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → 𝐴 Fn dom 𝐴)
109anim1i 591 . . . . . 6 (((Fun 𝐴 ∧ dom 𝐴 ∈ On) ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
11103impa 1278 . . . . 5 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
12 df-f 5930 . . . . 5 (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
1311, 12sylibr 224 . . . 4 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → 𝐴:dom 𝐴⟶{1𝑜, 2𝑜})
14 feq2 6065 . . . . 5 (𝑥 = dom 𝐴 → (𝐴:𝑥⟶{1𝑜, 2𝑜} ↔ 𝐴:dom 𝐴⟶{1𝑜, 2𝑜}))
1514rspcev 3340 . . . 4 ((dom 𝐴 ∈ On ∧ 𝐴:dom 𝐴⟶{1𝑜, 2𝑜}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})
165, 13, 15syl2anc 694 . . 3 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})
17 elno 31924 . . 3 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})
1816, 17sylibr 224 . 2 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → 𝐴 No )
194, 18impbii 199 1 (𝐴 No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∃wrex 2942   ⊆ 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:  elno3  31933  noextend  31944  noextendseq  31945  nosupno  31974
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