Theorem smofvon2 7987

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 7978	. . . 4 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
2	1	simp1bi 1141	. . 3 ⊢ (Smo 𝐹 → 𝐹:dom 𝐹⟶On)
3		ffvelrn 6843	. . . 4 ⊢ ((𝐹:dom 𝐹⟶On ∧ 𝐵 ∈ dom 𝐹) → (𝐹‘𝐵) ∈ On)
4	3	expcom 416	. . 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 6694	. . . 4 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
7		0elon 6238	. . . 4 ⊢ ∅ ∈ On
8	6, 7	eqeltrdi 2921	. . 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 184	1 ⊢ (Smo 𝐹 → (𝐹‘𝐵) ∈ On)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2110  ∀wral 3138  ∅c0 4290  dom cdm 5549  Ord word 6184  Oncon0 6185  ⟶wf 6345  ‘cfv 6349  Smo wsmo 7976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-tr 5165  df-id 5454  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-ord 6188  df-on 6189  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-smo 7977

This theorem is referenced by:  smo11  7995  smoord  7996  smoword  7997  smogt  7998