Theorem iunon 6428

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 4980	. . 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 5863	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
4		rnexg 4988	. . . 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 2229	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	fmpt 5784	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ On ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶On)
8		frn 5481	. . . 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 4579	. . . 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 2306	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ⊆ wss 3197  ∪ cuni 3887  ∪ ciun 3964   ↦ cmpt 4144  Oncon0 4453  ran crn 4719  ⟶wf 5313

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325

This theorem is referenced by:  rdgon  6530