Theorem smoel2 7460

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 5990	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2	1	eleq2d 2687	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
3	2	anbi1d 741	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))
4	3	biimprd 238	. . 3 ⊢ (𝐹 Fn 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵)))
5		smoel 7457	. . . 4 ⊢ ((Smo 𝐹 ∧ 𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹‘𝐵))
6	5	3expib 1268	. . 3 ⊢ (Smo 𝐹 → ((𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹‘𝐵)))
7	4, 6	sylan9 689	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹‘𝐵)))
8	7	imp 445	1 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) → (𝐹‘𝐶) ∈ (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  dom cdm 5114   Fn wfn 5883  ‘cfv 5888  Smo wsmo 7442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-tr 4753  df-ord 5726  df-iota 5851  df-fn 5891  df-fv 5896  df-smo 7443

This theorem is referenced by:  smo11  7461  smoord  7462  smogt  7464  cofsmo  9091