Theorem smofvon2 7995

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 7986	. . . 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 6851	. . . 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 6702	. . . 4 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
7		0elon 6246	. . . 4 ⊢ ∅ ∈ On
8	6, 7	eqeltrdi 2923	. . 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 2114  ∀wral 3140  ∅c0 4293  dom cdm 5557  Ord word 6192  Oncon0 6193  ⟶wf 6353  ‘cfv 6357  Smo wsmo 7984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-id 5462  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-ord 6196  df-on 6197  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-smo 7985

This theorem is referenced by:  smo11  8003  smoord  8004  smoword  8005  smogt  8006