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Theorem smoel2 7325
Description: A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smoel2 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐵𝐴𝐶𝐵)) → (𝐹𝐶) ∈ (𝐹𝐵))

Proof of Theorem smoel2
StepHypRef Expression
1 fndm 5890 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
21eleq2d 2673 . . . . 5 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
32anbi1d 737 . . . 4 (𝐹 Fn 𝐴 → ((𝐵 ∈ dom 𝐹𝐶𝐵) ↔ (𝐵𝐴𝐶𝐵)))
43biimprd 237 . . 3 (𝐹 Fn 𝐴 → ((𝐵𝐴𝐶𝐵) → (𝐵 ∈ dom 𝐹𝐶𝐵)))
5 smoel 7322 . . . 4 ((Smo 𝐹𝐵 ∈ dom 𝐹𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵))
653expib 1260 . . 3 (Smo 𝐹 → ((𝐵 ∈ dom 𝐹𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵)))
74, 6sylan9 687 . 2 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝐵𝐴𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵)))
87imp 444 1 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐵𝐴𝐶𝐵)) → (𝐹𝐶) ∈ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977  dom cdm 5028   Fn wfn 5785  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-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-tr 4676  df-ord 5629  df-iota 5754  df-fn 5793  df-fv 5798  df-smo 7308
This theorem is referenced by:  smo11  7326  smoord  7327  smogt  7329  cofsmo  8952
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