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Theorem issmo 8343
Description: Conditions for which 𝐴 is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.) Avoid ax-13 2372. (Revised by Gino Giotto, 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 6706 . . 3 (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On)
41, 3mpbir 230 . 2 𝐴:dom 𝐴⟶On
5 issmo.2 . . 3 Ord 𝐵
6 ordeq 6368 . . . 4 (dom 𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord 𝐵))
72, 6ax-mp 5 . . 3 (Ord dom 𝐴 ↔ Ord 𝐵)
85, 7mpbir 230 . 2 Ord dom 𝐴
92eleq2i 2826 . . . 4 (𝑥 ∈ dom 𝐴𝑥𝐵)
102eleq2i 2826 . . . 4 (𝑦 ∈ dom 𝐴𝑦𝐵)
11 issmo.3 . . . 4 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
129, 10, 11syl2anb 599 . . 3 ((𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
1312rgen2 3198 . 2 𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))
14 df-smo 8341 . 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 205  wa 397   = wceq 1542  wcel 2107  wral 3062  dom cdm 5675  Ord word 6360  Oncon0 6361  wf 6536  cfv 6540  Smo wsmo 8340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3954  df-ss 3964  df-uni 4908  df-tr 5265  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-fn 6543  df-f 6544  df-smo 8341
This theorem is referenced by:  iordsmo  8352  smobeth  10577
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