Theorem elno2 28068

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.

Step	Hyp	Ref	Expression
1		nofun 28063	. . 3 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		nodmon 28064	. . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
3		norn 28065	. . 3 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
4	1, 2, 3	3jca 1146	. 2 ⊢ (𝐴 ∈ No → (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
5		simp2 1155	. . . 4 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → dom 𝐴 ∈ On)
6		simpl 488	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → Fun 𝐴)
7	6	funfnd 6599	. . . . . . 7 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → 𝐴 Fn dom 𝐴)
8	7	anim1i 627	. . . . . 6 ⊢ (((Fun 𝐴 ∧ dom 𝐴 ∈ On) ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
9	8	3impa 1127	. . . . 5 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
10		df-f 6570	. . . . 5 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
11	9, 10	sylibr 237	. . . 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}))
13	12	rspcev 3580	. . . 4 ⊢ ((dom 𝐴 ∈ On ∧ 𝐴:dom 𝐴⟶{1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
14	5, 11, 13	syl2anc 596	. . 3 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
15		elno 28060	. . 3 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
16	14, 15	sylibr 237	. 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴 ∈ No)
17	4, 16	impbii 212	1 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  {cpr 4590  dom cdm 5658  ran crn 5659  Oncon0 6388  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562  1_oc1o 8507  2_oc2o 8508  Nocsur 28054

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-fun 6568  df-fn 6569  df-f 6570  df-no 28057

This theorem is used by:  elno3  28069  noextend  28080  noextendseq  28081  nosupno  28117  noinfno  28132  onnoxpg  44691