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Theorem issmo 8388
Description: Conditions for which 𝐴 is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) Avoid ax-13 2377. (Revised by GG, 19-May-2023.)
Hypotheses
Ref Expression
issmo.1 𝐴:𝐵⟶On
issmo.2 Ord 𝐵
issmo.3 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
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
StepHypRef Expression
1 issmo.1 . . 3 𝐴:𝐵⟶On
2 issmo.4 . . . 4 dom 𝐴 = 𝐵
32feq2i 6728 . . 3 (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On)
41, 3mpbir 231 . 2 𝐴:dom 𝐴⟶On
5 issmo.2 . . 3 Ord 𝐵
6 ordeq 6391 . . . 4 (dom 𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord 𝐵))
72, 6ax-mp 5 . . 3 (Ord dom 𝐴 ↔ Ord 𝐵)
85, 7mpbir 231 . 2 Ord dom 𝐴
92eleq2i 2833 . . . 4 (𝑥 ∈ dom 𝐴𝑥𝐵)
102eleq2i 2833 . . . 4 (𝑦 ∈ dom 𝐴𝑦𝐵)
11 issmo.3 . . . 4 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
129, 10, 11syl2anb 598 . . 3 ((𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
1312rgen2 3199 . 2 𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))
14 df-smo 8386 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
154, 8, 13, 14mpbir3an 1342 1 Smo 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  dom cdm 5685  Ord word 6383  Oncon0 6384  wf 6557  cfv 6561  Smo wsmo 8385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-fn 6564  df-f 6565  df-smo 8386
This theorem is referenced by:  iordsmo  8397  smobeth  10626
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