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Theorem smoel2 5974
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 5050 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
21eleq2d 2152 . . . . 5 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
32anbi1d 453 . . . 4 (𝐹 Fn 𝐴 → ((𝐵 ∈ dom 𝐹𝐶𝐵) ↔ (𝐵𝐴𝐶𝐵)))
43biimprd 156 . . 3 (𝐹 Fn 𝐴 → ((𝐵𝐴𝐶𝐵) → (𝐵 ∈ dom 𝐹𝐶𝐵)))
5 smoel 5971 . . . 4 ((Smo 𝐹𝐵 ∈ dom 𝐹𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵))
653expib 1142 . . 3 (Smo 𝐹 → ((𝐵 ∈ dom 𝐹𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵)))
74, 6sylan9 401 . 2 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝐵𝐴𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵)))
87imp 122 1 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐵𝐴𝐶𝐵)) → (𝐹𝐶) ∈ (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1434  dom cdm 4392   Fn wfn 4948  cfv 4953  Smo wsmo 5956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-tr 3897  df-iord 4150  df-iota 4918  df-fn 4956  df-fv 4961  df-smo 5957
This theorem is referenced by: (None)
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