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Theorem smoiun 6066
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 3734 . . 3 (𝑦 𝑥𝐴 (𝐵𝑥) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝑥))
2 smofvon 6064 . . . . 5 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐵𝐴) ∈ On)
3 smoel 6065 . . . . . 6 ((Smo 𝐵𝐴 ∈ dom 𝐵𝑥𝐴) → (𝐵𝑥) ∈ (𝐵𝐴))
433expia 1145 . . . . 5 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑥𝐴 → (𝐵𝑥) ∈ (𝐵𝐴)))
5 ontr1 4216 . . . . . 6 ((𝐵𝐴) ∈ On → ((𝑦 ∈ (𝐵𝑥) ∧ (𝐵𝑥) ∈ (𝐵𝐴)) → 𝑦 ∈ (𝐵𝐴)))
65expcomd 1375 . . . . 5 ((𝐵𝐴) ∈ On → ((𝐵𝑥) ∈ (𝐵𝐴) → (𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴))))
72, 4, 6sylsyld 57 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑥𝐴 → (𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴))))
87rexlimdv 2488 . . 3 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (∃𝑥𝐴 𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴)))
91, 8syl5bi 150 . 2 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑦 𝑥𝐴 (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴)))
109ssrdv 3031 1 ((Smo 𝐵𝐴 ∈ dom 𝐵) → 𝑥𝐴 (𝐵𝑥) ⊆ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1438  wrex 2360  wss 2999   ciun 3730  Oncon0 4190  dom cdm 4438  cfv 5015  Smo wsmo 6050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-smo 6051
This theorem is referenced by: (None)
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