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Theorem smoiun 7741
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 𝐵𝐴 ∈ dom 𝐵) → 𝑥𝐴 (𝐵𝑥) ⊆ (𝐵𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem smoiun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4757 . . 3 (𝑦 𝑥𝐴 (𝐵𝑥) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝑥))
2 smofvon 7739 . . . . 5 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐵𝐴) ∈ On)
3 smoel 7740 . . . . . 6 ((Smo 𝐵𝐴 ∈ dom 𝐵𝑥𝐴) → (𝐵𝑥) ∈ (𝐵𝐴))
433expia 1111 . . . . 5 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑥𝐴 → (𝐵𝑥) ∈ (𝐵𝐴)))
5 ontr1 6022 . . . . . 6 ((𝐵𝐴) ∈ On → ((𝑦 ∈ (𝐵𝑥) ∧ (𝐵𝑥) ∈ (𝐵𝐴)) → 𝑦 ∈ (𝐵𝐴)))
65expcomd 408 . . . . 5 ((𝐵𝐴) ∈ On → ((𝐵𝑥) ∈ (𝐵𝐴) → (𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴))))
72, 4, 6sylsyld 61 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑥𝐴 → (𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴))))
87rexlimdv 3211 . . 3 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (∃𝑥𝐴 𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴)))
91, 8syl5bi 234 . 2 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑦 𝑥𝐴 (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴)))
109ssrdv 3826 1 ((Smo 𝐵𝐴 ∈ dom 𝐵) → 𝑥𝐴 (𝐵𝑥) ⊆ (𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2106  wrex 3090  wss 3791   ciun 4753  dom cdm 5355  Oncon0 5976  cfv 6135  Smo wsmo 7725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-tr 4988  df-id 5261  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-ord 5979  df-on 5980  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-smo 7726
This theorem is referenced by: (None)
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