Theorem smoiun 8399

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 4999	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵‘𝑥))
2		smofvon 8397	. . . . 5 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐵‘𝐴) ∈ On)
3		smoel 8398	. . . . . 6 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐵‘𝑥) ∈ (𝐵‘𝐴))
4	3	3expia 1120	. . . . 5 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑥 ∈ 𝐴 → (𝐵‘𝑥) ∈ (𝐵‘𝐴)))
5		ontr1 6431	. . . . . 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 242	. 2 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴)))
10	9	ssrdv 4000	1 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ⊆ (𝐵‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2105  ∃wrex 3067   ⊆ wss 3962  ∪ ciun 4995  dom cdm 5688  Oncon0 6385  ‘cfv 6562  Smo wsmo 8383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-tr 5265  df-id 5582  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-ord 6388  df-on 6389  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-smo 8384

This theorem is referenced by: (None)