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Theorem elno2 31543
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 31538 . . 3 (𝐴 No → Fun 𝐴)
2 nodmon 31539 . . 3 (𝐴 No → dom 𝐴 ∈ On)
3 norn 31540 . . 3 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
41, 2, 33jca 1240 . 2 (𝐴 No → (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
5 simp2 1060 . . . 4 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → dom 𝐴 ∈ On)
6 simpl 473 . . . . . . . 8 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → Fun 𝐴)
7 eqidd 2622 . . . . . . . 8 ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → dom 𝐴 = dom 𝐴)
8 df-fn 5855 . . . . . . . 8 (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴))
96, 7, 8sylanbrc 697 . . . . . . 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 1256 . . . . 5 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
12 df-f 5856 . . . . 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 5989 . . . . 5 (𝑥 = dom 𝐴 → (𝐴:𝑥⟶{1𝑜, 2𝑜} ↔ 𝐴:dom 𝐴⟶{1𝑜, 2𝑜}))
1514rspcev 3298 . . . 4 ((dom 𝐴 ∈ On ∧ 𝐴:dom 𝐴⟶{1𝑜, 2𝑜}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})
165, 13, 15syl2anc 692 . . 3 ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})
17 elno 31535 . . 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 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  wss 3559  {cpr 4155  dom cdm 5079  ran crn 5080  Oncon0 5687  Fun wfun 5846   Fn wfn 5847  wf 5848  1𝑜c1o 7505  2𝑜c2o 7506   No csur 31529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-no 31532
This theorem is referenced by:  elno3  31544  noextend  31559  nosino  31610
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