Theorem smoel2 6304

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 5316	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2	1	eleq2d 2247	. . . . 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 6301	. . . 4 ⊢ ((Smo 𝐹 ∧ 𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹‘𝐵))
6	5	3expib 1206	. . 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 2148  dom cdm 4627   Fn wfn 5212  ‘cfv 5217  Smo wsmo 6286

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-tr 4103  df-iord 4367  df-iota 5179  df-fn 5220  df-fv 5225  df-smo 6287

This theorem is referenced by: (None)