Theorem issmo 5926

Description: Conditions for which 𝐴 is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)

Hypotheses

Ref	Expression
issmo.1	⊢ 𝐴:𝐵⟶On
issmo.2	⊢ Ord 𝐵
issmo.3	⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
issmo.4	⊢ dom 𝐴 = 𝐵

Assertion

Ref	Expression
issmo	⊢ Smo 𝐴

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem issmo

Step	Hyp	Ref	Expression
1		issmo.1	. . 3 ⊢ 𝐴:𝐵⟶On
2		issmo.4	. . . 4 ⊢ dom 𝐴 = 𝐵
3	2	feq2i 5060	. . 3 ⊢ (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On)
4	1, 3	mpbir 144	. 2 ⊢ 𝐴:dom 𝐴⟶On
5		issmo.2	. . 3 ⊢ Ord 𝐵
6		ordeq 4127	. . . 4 ⊢ (dom 𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord 𝐵))
7	2, 6	ax-mp 7	. . 3 ⊢ (Ord dom 𝐴 ↔ Ord 𝐵)
8	5, 7	mpbir 144	. 2 ⊢ Ord dom 𝐴
9	2	eleq2i 2145	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ 𝐵)
10	2	eleq2i 2145	. . . 4 ⊢ (𝑦 ∈ dom 𝐴 ↔ 𝑦 ∈ 𝐵)
11		issmo.3	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
12	9, 10, 11	syl2anb 285	. . 3 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ dom 𝐴) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
13	12	rgen2a 2417	. 2 ⊢ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))
14		df-smo 5924	. 2 ⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))))
15	4, 8, 13, 14	mpbir3an 1120	1 ⊢ Smo 𝐴

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348  Ord word 4117  Oncon0 4118  dom cdm 4363  ⟶wf 4918  ‘cfv 4922  Smo wsmo 5923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121  df-fn 4925  df-f 4926  df-smo 5924

This theorem is referenced by:  iordsmo  5935