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Theorem smoel2 8310
Description: A strictly monotone ordinal function preserves the membership 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 6606 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
21eleq2d 2820 . . . . 5 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
32anbi1d 631 . . . 4 (𝐹 Fn 𝐴 → ((𝐵 ∈ dom 𝐹𝐶𝐵) ↔ (𝐵𝐴𝐶𝐵)))
43biimprd 248 . . 3 (𝐹 Fn 𝐴 → ((𝐵𝐴𝐶𝐵) → (𝐵 ∈ dom 𝐹𝐶𝐵)))
5 smoel 8307 . . . 4 ((Smo 𝐹𝐵 ∈ dom 𝐹𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵))
653expib 1123 . . 3 (Smo 𝐹 → ((𝐵 ∈ dom 𝐹𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵)))
74, 6sylan9 509 . 2 ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝐵𝐴𝐶𝐵) → (𝐹𝐶) ∈ (𝐹𝐵)))
87imp 408 1 (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐵𝐴𝐶𝐵)) → (𝐹𝐶) ∈ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  dom cdm 5634   Fn wfn 6492  cfv 6497  Smo wsmo 8292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-tr 5224  df-ord 6321  df-iota 6449  df-fn 6500  df-fv 6505  df-smo 8293
This theorem is referenced by:  smo11  8311  smoord  8312  smogt  8314  cofsmo  10210
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