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Theorem iunon 5840
 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 ((A 𝑉 x A B On) → x A B On)
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   𝑉(x)

Proof of Theorem iunon
StepHypRef Expression
1 dfiun3g 4532 . . 3 (x A B On → x A B = ran (x AB))
21adantl 262 . 2 ((A 𝑉 x A B On) → x A B = ran (x AB))
3 mptexg 5329 . . . 4 (A 𝑉 → (x AB) V)
4 rnexg 4540 . . . 4 ((x AB) V → ran (x AB) V)
53, 4syl 14 . . 3 (A 𝑉 → ran (x AB) V)
6 eqid 2037 . . . . 5 (x AB) = (x AB)
76fmpt 5262 . . . 4 (x A B On ↔ (x AB):A⟶On)
8 frn 4995 . . . 4 ((x AB):A⟶On → ran (x AB) ⊆ On)
97, 8sylbi 114 . . 3 (x A B On → ran (x AB) ⊆ On)
10 ssonuni 4180 . . . 4 (ran (x AB) V → (ran (x AB) ⊆ On → ran (x AB) On))
1110imp 115 . . 3 ((ran (x AB) V ran (x AB) ⊆ On) → ran (x AB) On)
125, 9, 11syl2an 273 . 2 ((A 𝑉 x A B On) → ran (x AB) On)
132, 12eqeltrd 2111 1 ((A 𝑉 x A B On) → x A B On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ⊆ wss 2911  ∪ cuni 3571  ∪ ciun 3648   ↦ cmpt 3809  Oncon0 4066  ran crn 4289  ⟶wf 4841 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853 This theorem is referenced by:  rdgon  5913
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