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Theorem elno2 32347
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 32342 . . 3 (𝐴 No → Fun 𝐴)
2 nodmon 32343 . . 3 (𝐴 No → dom 𝐴 ∈ On)
3 norn 32344 . . 3 (𝐴 No → ran 𝐴 ⊆ {1o, 2o})
41, 2, 33jca 1164 . 2 (𝐴 No → (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
5 simp2 1173 . . . 4 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → dom 𝐴 ∈ On)
6 simpl 476 . . . . . . . 8 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → Fun 𝐴)
7 eqidd 2827 . . . . . . . 8 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → dom 𝐴 = dom 𝐴)
8 df-fn 6127 . . . . . . . 8 (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴))
96, 7, 8sylanbrc 580 . . . . . . 7 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → 𝐴 Fn dom 𝐴)
109anim1i 610 . . . . . 6 (((Fun 𝐴 ∧ dom 𝐴 ∈ On) ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
11103impa 1142 . . . . 5 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
12 df-f 6128 . . . . 5 (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
1311, 12sylibr 226 . . . 4 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴:dom 𝐴⟶{1o, 2o})
14 feq2 6261 . . . . 5 (𝑥 = dom 𝐴 → (𝐴:𝑥⟶{1o, 2o} ↔ 𝐴:dom 𝐴⟶{1o, 2o}))
1514rspcev 3527 . . . 4 ((dom 𝐴 ∈ On ∧ 𝐴:dom 𝐴⟶{1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
165, 13, 15syl2anc 581 . . 3 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
17 elno 32339 . . 3 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
1816, 17sylibr 226 . 2 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴 No )
194, 18impbii 201 1 (𝐴 No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wrex 3119  wss 3799  {cpr 4400  dom cdm 5343  ran crn 5344  Oncon0 5964  Fun wfun 6118   Fn wfn 6119  wf 6120  1oc1o 7820  2oc2o 7821   No csur 32333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-no 32336
This theorem is referenced by:  elno3  32348  noextend  32359  noextendseq  32360  nosupno  32389
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