Theorem issmo 8368

Description: Conditions for which 𝐴 is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) Avoid ax-13 2367. (Revised by Gino Giotto, 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 6714	. . 3 ⊢ (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On)
4	1, 3	mpbir 230	. 2 ⊢ 𝐴:dom 𝐴⟶On
5		issmo.2	. . 3 ⊢ Ord 𝐵
6		ordeq 6376	. . . 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 230	. 2 ⊢ Ord dom 𝐴
9	2	eleq2i 2821	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ 𝐵)
10	2	eleq2i 2821	. . . 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 597	. . 3 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ dom 𝐴) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
13	12	rgen2 3194	. 2 ⊢ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))
14		df-smo 8366	. 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 1339	1 ⊢ Smo 𝐴

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3058  dom cdm 5678  Ord word 6368  Oncon0 6369  ⟶wf 6544  ‘cfv 6548  Smo wsmo 8365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-v 3473  df-in 3954  df-ss 3964  df-uni 4909  df-tr 5266  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6372  df-fn 6551  df-f 6552  df-smo 8366

This theorem is referenced by:  iordsmo  8377  smobeth  10609