Theorem onovuni 8381

Description: A variant of onfununi 8380 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
onovuni.1	⊢ (Lim 𝑦 → (𝐴𝐹𝑦) = ∪ 𝑥 ∈ 𝑦 (𝐴𝐹𝑥))
onovuni.2	⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦))

Assertion

Ref	Expression
onovuni	⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇

Allowed substitution hint:   𝑇(𝑦)

Proof of Theorem onovuni

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onovuni.1	. . . 4 ⊢ (Lim 𝑦 → (𝐴𝐹𝑦) = ∪ 𝑥 ∈ 𝑦 (𝐴𝐹𝑥))
2		oveq2 7439	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧) = (𝐴𝐹𝑦))
3		eqid 2735	. . . . . 6 ⊢ (𝑧 ∈ V ↦ (𝐴𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐴𝐹𝑧))
4		ovex 7464	. . . . . 6 ⊢ (𝐴𝐹𝑦) ∈ V
5	2, 3, 4	fvmpt 7016	. . . . 5 ⊢ (𝑦 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦))
6	5	elv 3483	. . . 4 ⊢ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦)
7		oveq2 7439	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝐴𝐹𝑧) = (𝐴𝐹𝑥))
8		ovex 7464	. . . . . . . 8 ⊢ (𝐴𝐹𝑥) ∈ V
9	7, 3, 8	fvmpt 7016	. . . . . . 7 ⊢ (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
10	9	elv 3483	. . . . . 6 ⊢ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥)
11	10	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
12	11	iuneq2i 5018	. . . 4 ⊢ ∪ 𝑥 ∈ 𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = ∪ 𝑥 ∈ 𝑦 (𝐴𝐹𝑥)
13	1, 6, 12	3eqtr4g 2800	. . 3 ⊢ (Lim 𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = ∪ 𝑥 ∈ 𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
14		onovuni.2	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦))
15	14, 10, 6	3sstr4g 4041	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) ⊆ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦))
16	13, 15	onfununi 8380	. 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
17		uniexg 7759	. . . 4 ⊢ (𝑆 ∈ 𝑇 → ∪ 𝑆 ∈ V)
18		oveq2 7439	. . . . 5 ⊢ (𝑧 = ∪ 𝑆 → (𝐴𝐹𝑧) = (𝐴𝐹∪ 𝑆))
19		ovex 7464	. . . . 5 ⊢ (𝐴𝐹∪ 𝑆) ∈ V
20	18, 3, 19	fvmpt 7016	. . . 4 ⊢ (∪ 𝑆 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = (𝐴𝐹∪ 𝑆))
21	17, 20	syl 17	. . 3 ⊢ (𝑆 ∈ 𝑇 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = (𝐴𝐹∪ 𝑆))
22	21	3ad2ant1 1132	. 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = (𝐴𝐹∪ 𝑆))
23	10	a1i 11	. . . 4 ⊢ (𝑥 ∈ 𝑆 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
24	23	iuneq2i 5018	. . 3 ⊢ ∪ 𝑥 ∈ 𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥)
25	24	a1i 11	. 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∪ 𝑥 ∈ 𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥))
26	16, 22, 25	3eqtr3d 2783	1 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  Vcvv 3478   ⊆ wss 3963  ∅c0 4339  ∪ cuni 4912  ∪ ciun 4996   ↦ cmpt 5231  Oncon0 6386  Lim wlim 6387  ‘cfv 6563  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-lim 6391  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434

This theorem is referenced by:  onoviun  8382