Theorem smoel2 6356

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 5353	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2	1	eleq2d 2263	. . . . 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 6353	. . . 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 2164  dom cdm 4659   Fn wfn 5249  ‘cfv 5254  Smo wsmo 6338

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-tr 4128  df-iord 4397  df-iota 5215  df-fn 5257  df-fv 5262  df-smo 6339

This theorem is referenced by: (None)