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Theorem iunon 6185
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 4800 . . 3 (∀𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
21adantl 275 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
3 mptexg 5649 . . . 4 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
4 rnexg 4808 . . . 4 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
53, 4syl 14 . . 3 (𝐴𝑉 → ran (𝑥𝐴𝐵) ∈ V)
6 eqid 2140 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76fmpt 5574 . . . 4 (∀𝑥𝐴 𝐵 ∈ On ↔ (𝑥𝐴𝐵):𝐴⟶On)
8 frn 5285 . . . 4 ((𝑥𝐴𝐵):𝐴⟶On → ran (𝑥𝐴𝐵) ⊆ On)
97, 8sylbi 120 . . 3 (∀𝑥𝐴 𝐵 ∈ On → ran (𝑥𝐴𝐵) ⊆ On)
10 ssonuni 4408 . . . 4 (ran (𝑥𝐴𝐵) ∈ V → (ran (𝑥𝐴𝐵) ⊆ On → ran (𝑥𝐴𝐵) ∈ On))
1110imp 123 . . 3 ((ran (𝑥𝐴𝐵) ∈ V ∧ ran (𝑥𝐴𝐵) ⊆ On) → ran (𝑥𝐴𝐵) ∈ On)
125, 9, 11syl2an 287 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → ran (𝑥𝐴𝐵) ∈ On)
132, 12eqeltrd 2217 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → 𝑥𝐴 𝐵 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  wral 2417  Vcvv 2687  wss 3072   cuni 3740   ciun 3817  cmpt 3993  Oncon0 4289  ran crn 4544  wf 5123
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-id 4219  df-iord 4292  df-on 4294  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135
This theorem is referenced by:  rdgon  6287
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