Theorem issmo 8384

Description: Conditions for which 𝐴 is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) Avoid ax-13 2404. (Revised by GG, 19-May-2023.)

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 6723	. . 3 ⊢ (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On)
4	1, 3	mpbir 234	. 2 ⊢ 𝐴:dom 𝐴⟶On
5		issmo.2	. . 3 ⊢ Ord 𝐵
6		ordeq 6387	. . . 4 ⊢ (dom 𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord 𝐵))
7	2, 6	ax-mp 5	. . 3 ⊢ (Ord dom 𝐴 ↔ Ord 𝐵)
8	5, 7	mpbir 234	. 2 ⊢ Ord dom 𝐴
9	2	eleq2i 2856	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ 𝐵)
10	2	eleq2i 2856	. . . 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 610	. . 3 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ dom 𝐴) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
13	12	rgen2 3206	. 2 ⊢ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))
14		df-smo 8382	. 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 1360	1 ⊢ Smo 𝐴

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  dom cdm 5655  Ord word 6379  Oncon0 6380  ⟶wf 6554  ‘cfv 6558  Smo wsmo 8381

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-uni 4871  df-tr 5216  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6383  df-fn 6561  df-f 6562  df-smo 8382

This theorem is used by:  iordsmo  8393  smobeth  10698  j00smo  36005