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Theorem smoel2 5948
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 5025 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
21eleq2d 2123 . . . . 5 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
32anbi1d 446 . . . 4 (𝐹 Fn 𝐴 → ((𝐵 ∈ dom 𝐹𝐶𝐵) ↔ (𝐵𝐴𝐶𝐵)))
43biimprd 151 . . 3 (𝐹 Fn 𝐴 → ((𝐵𝐴𝐶𝐵) → (𝐵 ∈ dom 𝐹𝐶𝐵)))
5 smoel 5945 . . . 4 ((Smo 𝐹𝐵 ∈ dom 𝐹𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵))
653expib 1118 . . 3 (Smo 𝐹 → ((𝐵 ∈ dom 𝐹𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵)))
74, 6sylan9 395 . 2 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝐵𝐴𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵)))
87imp 119 1 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐵𝐴𝐶𝐵)) → (𝐹𝐶) ∈ (𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  dom cdm 4372   Fn wfn 4924  cfv 4929  Smo wsmo 5930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-tr 3882  df-iord 4130  df-iota 4894  df-fn 4932  df-fv 4937  df-smo 5931
This theorem is referenced by: (None)
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