Theorem smofvon2 8396

Description: The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)

Assertion

Ref	Expression
smofvon2	⊢ (Smo 𝐹 → (𝐹‘𝐵) ∈ On)

Proof of Theorem smofvon2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsmo2 8387	. . . 4 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
2	1	simp1bi 1146	. . 3 ⊢ (Smo 𝐹 → 𝐹:dom 𝐹⟶On)
3		ffvelcdm 7101	. . . 4 ⊢ ((𝐹:dom 𝐹⟶On ∧ 𝐵 ∈ dom 𝐹) → (𝐹‘𝐵) ∈ On)
4	3	expcom 413	. . 3 ⊢ (𝐵 ∈ dom 𝐹 → (𝐹:dom 𝐹⟶On → (𝐹‘𝐵) ∈ On))
5	2, 4	syl5 34	. 2 ⊢ (𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹‘𝐵) ∈ On))
6		ndmfv 6941	. . . 4 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
7		0elon 6438	. . . 4 ⊢ ∅ ∈ On
8	6, 7	eqeltrdi 2849	. . 3 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) ∈ On)
9	8	a1d 25	. 2 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (Smo 𝐹 → (𝐹‘𝐵) ∈ On))
10	5, 9	pm2.61i 182	1 ⊢ (Smo 𝐹 → (𝐹‘𝐵) ∈ On)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2108  ∀wral 3061  ∅c0 4333  dom cdm 5685  Ord word 6383  Oncon0 6384  ⟶wf 6557  ‘cfv 6561  Smo wsmo 8385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-smo 8386

This theorem is referenced by:  smo11  8404  smoord  8405  smoword  8406  smogt  8407