Theorem elno2 27636

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 27631	. . 3 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		nodmon 27632	. . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
3		norn 27633	. . 3 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
4	1, 2, 3	3jca 1134	. 2 ⊢ (𝐴 ∈ No → (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))
5		simp2 1143	. . . 4 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → dom 𝐴 ∈ On)
6		simpl 483	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → Fun 𝐴)
7	6	funfnd 6516	. . . . . . 7 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → 𝐴 Fn dom 𝐴)
8	7	anim1i 621	. . . . . 6 ⊢ (((Fun 𝐴 ∧ dom 𝐴 ∈ On) ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
9	8	3impa 1115	. . . . 5 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
10		df-f 6489	. . . . 5 ⊢ (𝐴:dom 𝐴⟶{1o, 2o} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}))
11	9, 10	sylibr 235	. . . 4 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴:dom 𝐴⟶{1o, 2o})
12		feq2 6634	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴:𝑥⟶{1o, 2o} ↔ 𝐴:dom 𝐴⟶{1o, 2o}))
13	12	rspcev 3560	. . . 4 ⊢ ((dom 𝐴 ∈ On ∧ 𝐴:dom 𝐴⟶{1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
14	5, 11, 13	syl2anc 590	. . 3 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
15		elno 27627	. . 3 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
16	14, 15	sylibr 235	. 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}) → 𝐴 ∈ No )
17	4, 16	impbii 210	1 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1o, 2o}))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119  ∃wrex 3063   ⊆ wss 3883  {cpr 4557  dom cdm 5618  ran crn 5619  Oncon0 6310  Fun wfun 6479   Fn wfn 6480  ⟶wf 6481  1_oc1o 8388  2_oc2o 8389   No csur 27621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-fun 6487  df-fn 6488  df-f 6489  df-no 27624

This theorem is referenced by:  elno3  27637  noextend  27648  noextendseq  27649  nosupno  27685  noinfno  27700  onnoxpg  43873