Theorem smoiun 8382

Description: The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)

Assertion

Ref	Expression
smoiun	⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ⊆ (𝐵‘𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem smoiun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 5000	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵‘𝑥))
2		smofvon 8380	. . . . 5 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐵‘𝐴) ∈ On)
3		smoel 8381	. . . . . 6 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐵‘𝑥) ∈ (𝐵‘𝐴))
4	3	3expia 1119	. . . . 5 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑥 ∈ 𝐴 → (𝐵‘𝑥) ∈ (𝐵‘𝐴)))
5		ontr1 6415	. . . . . 6 ⊢ ((𝐵‘𝐴) ∈ On → ((𝑦 ∈ (𝐵‘𝑥) ∧ (𝐵‘𝑥) ∈ (𝐵‘𝐴)) → 𝑦 ∈ (𝐵‘𝐴)))
6	5	expcomd 416	. . . . 5 ⊢ ((𝐵‘𝐴) ∈ On → ((𝐵‘𝑥) ∈ (𝐵‘𝐴) → (𝑦 ∈ (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴))))
7	2, 4, 6	sylsyld 61	. . . 4 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑥 ∈ 𝐴 → (𝑦 ∈ (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴))))
8	7	rexlimdv 3150	. . 3 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴)))
9	1, 8	biimtrid 241	. 2 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴)))
10	9	ssrdv 3986	1 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ⊆ (𝐵‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2099  ∃wrex 3067   ⊆ wss 3947  ∪ ciun 4996  dom cdm 5678  Oncon0 6369  ‘cfv 6548  Smo wsmo 8366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-tr 5266  df-id 5576  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-ord 6372  df-on 6373  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-smo 8367

This theorem is referenced by: (None)