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Theorem smoel2 5859
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) (B A 𝐶 B)) → (𝐹𝐶) (𝐹B))

Proof of Theorem smoel2
StepHypRef Expression
1 fndm 4941 . . . . . 6 (𝐹 Fn A → dom 𝐹 = A)
21eleq2d 2104 . . . . 5 (𝐹 Fn A → (B dom 𝐹B A))
32anbi1d 438 . . . 4 (𝐹 Fn A → ((B dom 𝐹 𝐶 B) ↔ (B A 𝐶 B)))
43biimprd 147 . . 3 (𝐹 Fn A → ((B A 𝐶 B) → (B dom 𝐹 𝐶 B)))
5 smoel 5856 . . . 4 ((Smo 𝐹 B dom 𝐹 𝐶 B) → (𝐹𝐶) (𝐹B))
653expib 1106 . . 3 (Smo 𝐹 → ((B dom 𝐹 𝐶 B) → (𝐹𝐶) (𝐹B)))
74, 6sylan9 389 . 2 ((𝐹 Fn A Smo 𝐹) → ((B A 𝐶 B) → (𝐹𝐶) (𝐹B)))
87imp 115 1 (((𝐹 Fn A Smo 𝐹) (B A 𝐶 B)) → (𝐹𝐶) (𝐹B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  dom cdm 4288   Fn wfn 4840  cfv 4845  Smo wsmo 5841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-tr 3846  df-iord 4069  df-iota 4810  df-fn 4848  df-fv 4853  df-smo 5842
This theorem is referenced by: (None)
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