Theorem elno2 27871

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 27866	. . 3 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		nodmon 27867	. . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
3		norn 27868	. . 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 6571	. . . . . . 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 6544	. . . . 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 6688	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴:𝑥⟶{1o, 2o} ↔ 𝐴:dom 𝐴⟶{1o, 2o}))
13	12	rspcev 3583	. . . 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 27863	. . 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 2146  ∃wrex 3091   ⊆ wss 3906  {cpr 4593  dom cdm 5663  ran crn 5664  Oncon0 6364  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  1_oc1o 8452  2_oc2o 8453   No csur 27857

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-fun 6542  df-fn 6543  df-f 6544  df-no 27860

This theorem is used by:  elno3  27872  noextend  27883  noextendseq  27884  nosupno  27920  noinfno  27935  onnoxpg  44215