Theorem elno2 27573

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 27568	. . 3 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		nodmon 27569	. . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
3		norn 27570	. . 3 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
4	1, 2, 3	3jca 1128	. 2 ⊢ (𝐴 ∈ No → (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
5		simp2 1137	. . . 4 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → dom 𝐴 ∈ On)
6		simpl 482	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → Fun 𝐴)
7	6	funfnd 6550	. . . . . . 7 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → 𝐴 Fn dom 𝐴)
8	7	anim1i 615	. . . . . 6 ⊢ (((Fun 𝐴 ∧ dom 𝐴 ∈ On) ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
9	8	3impa 1109	. . . . 5 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
10		df-f 6518	. . . . 5 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
11	9, 10	sylibr 234	. . . 4 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴:dom 𝐴⟶{1o, 2o})
12		feq2 6670	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴:𝑥⟶{1o, 2o} ↔ 𝐴:dom 𝐴⟶{1o, 2o}))
13	12	rspcev 3591	. . . 4 ⊢ ((dom 𝐴 ∈ On ∧ 𝐴:dom 𝐴⟶{1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
14	5, 11, 13	syl2anc 584	. . 3 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
15		elno 27564	. . 3 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
16	14, 15	sylibr 234	. 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴 ∈ No )
17	4, 16	impbii 209	1 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109  ∃wrex 3054   ⊆ wss 3917  {cpr 4594  dom cdm 5641  ran crn 5642  Oncon0 6335  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  1_oc1o 8430  2_oc2o 8431   No csur 27558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-fun 6516  df-fn 6517  df-f 6518  df-no 27561

This theorem is referenced by:  elno3  27574  noextend  27585  noextendseq  27586  nosupno  27622  noinfno  27637  onnog  43425