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Theorem issmo 7614
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 ((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 6198 . . 3 (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On)
41, 3mpbir 221 . 2 𝐴:dom 𝐴⟶On
5 issmo.2 . . 3 Ord 𝐵
6 ordeq 5891 . . . 4 (dom 𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord 𝐵))
72, 6ax-mp 5 . . 3 (Ord dom 𝐴 ↔ Ord 𝐵)
85, 7mpbir 221 . 2 Ord dom 𝐴
92eleq2i 2831 . . . 4 (𝑥 ∈ dom 𝐴𝑥𝐵)
102eleq2i 2831 . . . 4 (𝑦 ∈ dom 𝐴𝑦𝐵)
11 issmo.3 . . . 4 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
129, 10, 11syl2anb 497 . . 3 ((𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
1312rgen2a 3115 . 2 𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))
14 df-smo 7612 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
154, 8, 13, 14mpbir3an 1427 1 Smo 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  dom cdm 5266  Ord word 5883  Oncon0 5884  wf 6045  cfv 6049  Smo wsmo 7611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-in 3722  df-ss 3729  df-uni 4589  df-tr 4905  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-fn 6052  df-f 6053  df-smo 7612
This theorem is referenced by:  iordsmo  7623  smobeth  9600
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