Theorem elno2 31807

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 31802	. . 3 ⊢ (𝐴 ∈ No → Fun 𝐴)
2		nodmon 31803	. . 3 ⊢ (𝐴 ∈ No → dom 𝐴 ∈ On)
3		norn 31804	. . 3 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
4	1, 2, 3	3jca 1242	. 2 ⊢ (𝐴 ∈ No → (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
5		simp2 1062	. . . 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 2623	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → dom 𝐴 = dom 𝐴)
8		df-fn 5891	. . . . . . . 8 ⊢ (𝐴 Fn dom 𝐴 ↔ (Fun 𝐴 ∧ dom 𝐴 = dom 𝐴))
9	6, 7, 8	sylanbrc 698	. . . . . . 7 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On) → 𝐴 Fn dom 𝐴)
10	9	anim1i 592	. . . . . 6 ⊢ (((Fun 𝐴 ∧ dom 𝐴 ∈ On) ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
11	10	3impa 1259	. . . . 5 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
12		df-f 5892	. . . . 5 ⊢ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))
13	11, 12	sylibr 224	. . . 4 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → 𝐴:dom 𝐴⟶{1𝑜, 2𝑜})
14		feq2 6027	. . . . 5 ⊢ (𝑥 = dom 𝐴 → (𝐴:𝑥⟶{1𝑜, 2𝑜} ↔ 𝐴:dom 𝐴⟶{1𝑜, 2𝑜}))
15	14	rspcev 3309	. . . 4 ⊢ ((dom 𝐴 ∈ On ∧ 𝐴:dom 𝐴⟶{1𝑜, 2𝑜}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})
16	5, 13, 15	syl2anc 693	. . 3 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})
17		elno 31799	. . 3 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})
18	16, 17	sylibr 224	. 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → 𝐴 ∈ No )
19	4, 18	impbii 199	1 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ⊆ wss 3574  {cpr 4179  dom cdm 5114  ran crn 5115  Oncon0 5723  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  1_𝑜c1o 7553  2_𝑜c2o 7554   No csur 31793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-no 31796

This theorem is referenced by:  elno3  31808  noextend  31819  noextendseq  31820  nosupno  31849