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Theorem iunon 8335
Description: The indexed union of a set of ordinal numbers 𝐵(𝑥) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
iunon ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → 𝑥𝐴 𝐵 ∈ On)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem iunon
StepHypRef Expression
1 dfiun3g 5954 . . 3 (∀𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
21adantl 481 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
3 mptexg 7215 . . . 4 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
4 rnexg 7889 . . . 4 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
53, 4syl 17 . . 3 (𝐴𝑉 → ran (𝑥𝐴𝐵) ∈ V)
6 eqid 2724 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76rnmptss 7115 . . 3 (∀𝑥𝐴 𝐵 ∈ On → ran (𝑥𝐴𝐵) ⊆ On)
8 ssonuni 7761 . . . 4 (ran (𝑥𝐴𝐵) ∈ V → (ran (𝑥𝐴𝐵) ⊆ On → ran (𝑥𝐴𝐵) ∈ On))
98imp 406 . . 3 ((ran (𝑥𝐴𝐵) ∈ V ∧ ran (𝑥𝐴𝐵) ⊆ On) → ran (𝑥𝐴𝐵) ∈ On)
105, 7, 9syl2an 595 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → ran (𝑥𝐴𝐵) ∈ On)
112, 10eqeltrd 2825 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → 𝑥𝐴 𝐵 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3053  Vcvv 3466  wss 3941   cuni 4900   ciun 4988  cmpt 5222  ran crn 5668  Oncon0 6355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542
This theorem is referenced by:  oacl  8531  omcl  8532  oecl  8533  rankuni2b  9845  rankval4  9859  alephon  10061  cfsmolem  10262  hsmexlem5  10422  inar1  10767  ofoafg  42654
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