Theorem issmo 6549

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 5522	. . 3 ⊢ (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On)
4	1, 3	mpbir 146	. 2 ⊢ 𝐴:dom 𝐴⟶On
5		issmo.2	. . 3 ⊢ Ord 𝐵
6		ordeq 4512	. . . 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 146	. 2 ⊢ Ord dom 𝐴
9	2	eleq2i 2305	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ 𝐵)
10	2	eleq2i 2305	. . . 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 291	. . 3 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 𝑦 ∈ dom 𝐴) → (𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))
13	12	rgen2a 2604	. 2 ⊢ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦))
14		df-smo 6547	. 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 1210	1 ⊢ Smo 𝐴

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  Ord word 4502  Oncon0 4503  dom cdm 4769  ⟶wf 5368  ‘cfv 5372  Smo wsmo 6546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-in 3226  df-ss 3233  df-uni 3931  df-tr 4225  df-iord 4506  df-fn 5375  df-f 5376  df-smo 6547

This theorem is referenced by:  iordsmo  6558