Theorem smoel2 8349

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) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) → (𝐹‘𝐶) ∈ (𝐹‘𝐵))

Proof of Theorem smoel2

Step	Hyp	Ref	Expression
1		fndm 6639	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2	1	eleq2d 2855	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
3	2	anbi1d 642	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))
4	3	biimprd 251	. . 3 ⊢ (𝐹 Fn 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵)))
5		smoel 8346	. . . 4 ⊢ ((Smo 𝐹 ∧ 𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹‘𝐵))
6	5	3expib 1138	. . 3 ⊢ (Smo 𝐹 → ((𝐵 ∈ dom 𝐹 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹‘𝐵)))
7	4, 6	sylan9 516	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹‘𝐵)))
8	7	imp 411	1 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) → (𝐹‘𝐶) ∈ (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149  dom cdm 5662   Fn wfn 6532  ‘cfv 6537  Smo wsmo 8331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-tr 5223  df-ord 6364  df-iota 6493  df-fn 6540  df-fv 6545  df-smo 8332

This theorem is referenced by:  smo11  8350  smoord  8351  smogt  8353  cofsmo  10252