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Theorem smoel2 5828
 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 4912 . . . . . 6 (𝐹 Fn A → dom 𝐹 = A)
21eleq2d 2080 . . . . 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 5825 . . . 4 ((Smo 𝐹 B dom 𝐹 𝐶 B) → (𝐹𝐶) (𝐹B))
653expib 1088 . . 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 1366  dom cdm 4260   Fn wfn 4812  ‘cfv 4817  Smo wsmo 5810 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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995 This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-tr 3818  df-iord 4041  df-iota 4782  df-fn 4820  df-fv 4825  df-smo 5811 This theorem is referenced by: (None)
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