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Theorem iunon 5860
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 4550 . . 3 (∀𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
21adantl 262 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
3 mptexg 5347 . . . 4 (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
4 rnexg 4558 . . . 4 ((𝑥𝐴𝐵) ∈ V → ran (𝑥𝐴𝐵) ∈ V)
53, 4syl 14 . . 3 (𝐴𝑉 → ran (𝑥𝐴𝐵) ∈ V)
6 eqid 2040 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76fmpt 5280 . . . 4 (∀𝑥𝐴 𝐵 ∈ On ↔ (𝑥𝐴𝐵):𝐴⟶On)
8 frn 5013 . . . 4 ((𝑥𝐴𝐵):𝐴⟶On → ran (𝑥𝐴𝐵) ⊆ On)
97, 8sylbi 114 . . 3 (∀𝑥𝐴 𝐵 ∈ On → ran (𝑥𝐴𝐵) ⊆ On)
10 ssonuni 4185 . . . 4 (ran (𝑥𝐴𝐵) ∈ V → (ran (𝑥𝐴𝐵) ⊆ On → ran (𝑥𝐴𝐵) ∈ On))
1110imp 115 . . 3 ((ran (𝑥𝐴𝐵) ∈ V ∧ ran (𝑥𝐴𝐵) ⊆ On) → ran (𝑥𝐴𝐵) ∈ On)
125, 9, 11syl2an 273 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → ran (𝑥𝐴𝐵) ∈ On)
132, 12eqeltrd 2114 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ On) → 𝑥𝐴 𝐵 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wcel 1393  wral 2303  Vcvv 2554  wss 2914   cuni 3576   ciun 3653  cmpt 3814  Oncon0 4071  ran crn 4307  wf 4859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3868  ax-sep 3871  ax-pow 3923  ax-pr 3940  ax-un 4141
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-iun 3655  df-br 3761  df-opab 3815  df-mpt 3816  df-tr 3851  df-id 4026  df-iord 4074  df-on 4076  df-xp 4312  df-rel 4313  df-cnv 4314  df-co 4315  df-dm 4316  df-rn 4317  df-res 4318  df-ima 4319  df-iota 4828  df-fun 4865  df-fn 4866  df-f 4867  df-f1 4868  df-fo 4869  df-f1o 4870  df-fv 4871
This theorem is referenced by:  rdgon  5934
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