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Theorem smoiun 6164
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 3785 . . 3 (𝑦 𝑥𝐴 (𝐵𝑥) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝑥))
2 smofvon 6162 . . . . 5 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝐵𝐴) ∈ On)
3 smoel 6163 . . . . . 6 ((Smo 𝐵𝐴 ∈ dom 𝐵𝑥𝐴) → (𝐵𝑥) ∈ (𝐵𝐴))
433expia 1166 . . . . 5 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑥𝐴 → (𝐵𝑥) ∈ (𝐵𝐴)))
5 ontr1 4279 . . . . . 6 ((𝐵𝐴) ∈ On → ((𝑦 ∈ (𝐵𝑥) ∧ (𝐵𝑥) ∈ (𝐵𝐴)) → 𝑦 ∈ (𝐵𝐴)))
65expcomd 1400 . . . . 5 ((𝐵𝐴) ∈ On → ((𝐵𝑥) ∈ (𝐵𝐴) → (𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴))))
72, 4, 6sylsyld 58 . . . 4 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑥𝐴 → (𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴))))
87rexlimdv 2523 . . 3 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (∃𝑥𝐴 𝑦 ∈ (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴)))
91, 8syl5bi 151 . 2 ((Smo 𝐵𝐴 ∈ dom 𝐵) → (𝑦 𝑥𝐴 (𝐵𝑥) → 𝑦 ∈ (𝐵𝐴)))
109ssrdv 3071 1 ((Smo 𝐵𝐴 ∈ dom 𝐵) → 𝑥𝐴 (𝐵𝑥) ⊆ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1463  wrex 2392  wss 3039   ciun 3781  Oncon0 4253  dom cdm 4507  cfv 5091  Smo wsmo 6148
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-smo 6149
This theorem is referenced by: (None)
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