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Theorem smoiun 6280
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 3877 . . 3 (𝑦 𝑥𝐴 (𝐵𝑥) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝑥))
2 smofvon 6278 . . . . 5 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐵𝐴) ∈ On)
3 smoel 6279 . . . . . 6 ((Smo 𝐵𝐴 ∈ dom 𝐵𝑥𝐴) → (𝐵𝑥) ∈ (𝐵𝐴))
433expia 1200 . . . . 5 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑥𝐴 → (𝐵𝑥) ∈ (𝐵𝐴)))
5 ontr1 4374 . . . . . 6 ((𝐵𝐴) ∈ On → ((𝑦 ∈ (𝐵𝑥) ∧ (𝐵𝑥) ∈ (𝐵𝐴)) → 𝑦 ∈ (𝐵𝐴)))
65expcomd 1434 . . . . 5 ((𝐵𝐴) ∈ On → ((𝐵𝑥) ∈ (𝐵𝐴) → (𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴))))
72, 4, 6sylsyld 58 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑥𝐴 → (𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴))))
87rexlimdv 2586 . . 3 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (∃𝑥𝐴 𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴)))
91, 8syl5bi 151 . 2 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑦 𝑥𝐴 (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴)))
109ssrdv 3153 1 ((Smo 𝐵𝐴 ∈ dom 𝐵) → 𝑥𝐴 (𝐵𝑥) ⊆ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2141  wrex 2449  wss 3121   ciun 3873  Oncon0 4348  dom cdm 4611  cfv 5198  Smo wsmo 6264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-smo 6265
This theorem is referenced by: (None)
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