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Theorem issmo 7310
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 5936 . . 3 (𝐴:dom 𝐴⟶On ↔ 𝐴:𝐵⟶On)
41, 3mpbir 220 . 2 𝐴:dom 𝐴⟶On
5 issmo.2 . . 3 Ord 𝐵
6 ordeq 5633 . . . 4 (dom 𝐴 = 𝐵 → (Ord dom 𝐴 ↔ Ord 𝐵))
72, 6ax-mp 5 . . 3 (Ord dom 𝐴 ↔ Ord 𝐵)
85, 7mpbir 220 . 2 Ord dom 𝐴
92eleq2i 2680 . . . 4 (𝑥 ∈ dom 𝐴𝑥𝐵)
102eleq2i 2680 . . . 4 (𝑦 ∈ dom 𝐴𝑦𝐵)
11 issmo.3 . . . 4 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
129, 10, 11syl2anb 495 . . 3 ((𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴) → (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
1312rgen2a 2960 . 2 𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))
14 df-smo 7308 . 2 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
154, 8, 13, 14mpbir3an 1237 1 Smo 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  dom cdm 5028  Ord word 5625  Oncon0 5626  wf 5786  cfv 5790  Smo wsmo 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-in 3547  df-ss 3554  df-uni 4368  df-tr 4676  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-fn 5793  df-f 5794  df-smo 7308
This theorem is referenced by:  iordsmo  7319  smobeth  9265
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