Theorem smoiun 8000

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 4925	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵‘𝑥))
2		smofvon 7998	. . . . 5 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐵‘𝐴) ∈ On)
3		smoel 7999	. . . . . 6 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐵‘𝑥) ∈ (𝐵‘𝐴))
4	3	3expia 1117	. . . . 5 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑥 ∈ 𝐴 → (𝐵‘𝑥) ∈ (𝐵‘𝐴)))
5		ontr1 6239	. . . . . 6 ⊢ ((𝐵‘𝐴) ∈ On → ((𝑦 ∈ (𝐵‘𝑥) ∧ (𝐵‘𝑥) ∈ (𝐵‘𝐴)) → 𝑦 ∈ (𝐵‘𝐴)))
6	5	expcomd 419	. . . . 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 3285	. . 3 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴)))
9	1, 8	syl5bi 244	. 2 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴)))
10	9	ssrdv 3975	1 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ⊆ (𝐵‘𝐴))

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  ∃wrex 3141   ⊆ wss 3938  ∪ ciun 4921  dom cdm 5557  Oncon0 6193  ‘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-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-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-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  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: (None)