Theorem iunon 6515

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

Step	Hyp	Ref	Expression
1		dfiun3g 5014	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
2	1	adantl 277	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵))
3		mptexg 5911	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
4		rnexg 5022	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
5	3, 4	syl 14	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
6		eqid 2232	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	fmpt 5827	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶On)
8		frn 5517	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶On → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On)
9	7, 8	sylbi 121	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On)
10		ssonuni 4610	. . . 4 ⊢ (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V → (ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On))
11	10	imp 124	. . 3 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ On) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On)
12	5, 9, 11	syl2an 289	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ On) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ On)
13	2, 12	eqeltrd 2309	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ∀wral 2520  Vcvv 2813   ⊆ wss 3211  ∪ cuni 3914  ∪ ciun 3991   ↦ cmpt 4171  Oncon0 4484  ran crn 4750  ⟶wf 5348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360

This theorem is referenced by:  rdgon  6617