MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunon Structured version   Visualization version   GIF version

Theorem iunon 7481
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 5410 . . 3 (∀𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
21adantl 481 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
3 mptexg 6525 . . . 4 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
4 rnexg 7140 . . . 4 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
53, 4syl 17 . . 3 (𝐴𝑉 → ran (𝑥𝐴𝐵) ∈ V)
6 eqid 2651 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76rnmptss 6432 . . 3 (∀𝑥𝐴 𝐵 ∈ On → ran (𝑥𝐴𝐵) ⊆ On)
8 ssonuni 7028 . . . 4 (ran (𝑥𝐴𝐵) ∈ V → (ran (𝑥𝐴𝐵) ⊆ On → ran (𝑥𝐴𝐵) ∈ On))
98imp 444 . . 3 ((ran (𝑥𝐴𝐵) ∈ V ∧ ran (𝑥𝐴𝐵) ⊆ On) → ran (𝑥𝐴𝐵) ∈ On)
105, 7, 9syl2an 493 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → ran (𝑥𝐴𝐵) ∈ On)
112, 10eqeltrd 2730 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → 𝑥𝐴 𝐵 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607   cuni 4468   ciun 4552  cmpt 4762  ran crn 5144  Oncon0 5761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  oacl  7660  omcl  7661  oecl  7662  rankuni2b  8754  rankval4  8768  alephon  8930  cfsmolem  9130  hsmexlem5  9290  inar1  9635
  Copyright terms: Public domain W3C validator