Theorem smoel2 6388

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)) → (𝐹‘𝐶) ∈ (𝐹‘𝐵))

Proof of Theorem smoel2

Step	Hyp	Ref	Expression
1		fndm 5372	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2	1	eleq2d 2274	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
3	2	anbi1d 465	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))
4	3	biimprd 158	. . 3 ⊢ (𝐹 Fn 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵)))
5		smoel 6385	. . . 4 ⊢ ((Smo 𝐹 ∧ 𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹‘𝐵))
6	5	3expib 1208	. . 3 ⊢ (Smo 𝐹 → ((𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹‘𝐵)))
7	4, 6	sylan9 409	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹‘𝐵)))
8	7	imp 124	1 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) → (𝐹‘𝐶) ∈ (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2175  dom cdm 4674   Fn wfn 5265  ‘cfv 5270  Smo wsmo 6370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-tr 4142  df-iord 4412  df-iota 5231  df-fn 5273  df-fv 5278  df-smo 6371

This theorem is referenced by: (None)