Theorem onovuni 8357

Description: A variant of onfununi 8356 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 7432	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧) = (𝐴𝐹𝑦))
3		eqid 2760	. . . . . 6 ⊢ (𝑧 ∈ V ↦ (𝐴𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐴𝐹𝑧))
4		ovex 7457	. . . . . 6 ⊢ (𝐴𝐹𝑦) ∈ V
5	2, 3, 4	fvmpt 6998	. . . . 5 ⊢ (𝑦 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦))
6	5	elv 3455	. . . 4 ⊢ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦)
7		oveq2 7432	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝐴𝐹𝑧) = (𝐴𝐹𝑥))
8		ovex 7457	. . . . . . . 8 ⊢ (𝐴𝐹𝑥) ∈ V
9	7, 3, 8	fvmpt 6998	. . . . . . 7 ⊢ (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
10	9	elv 3455	. . . . . 6 ⊢ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥)
11	10	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
12	11	iuneq2i 4972	. . . 4 ⊢ ∪ 𝑥 ∈ 𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = ∪ 𝑥 ∈ 𝑦 (𝐴𝐹𝑥)
13	1, 6, 12	3eqtr4g 2820	. . 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 3962	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) ⊆ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦))
16	13, 15	onfununi 8356	. 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
17		uniexg 7761	. . . 4 ⊢ (𝑆 ∈ 𝑇 → ∪ 𝑆 ∈ V)
18		oveq2 7432	. . . . 5 ⊢ (𝑧 = ∪ 𝑆 → (𝐴𝐹𝑧) = (𝐴𝐹∪ 𝑆))
19		ovex 7457	. . . . 5 ⊢ (𝐴𝐹∪ 𝑆) ∈ V
20	18, 3, 19	fvmpt 6998	. . . 4 ⊢ (∪ 𝑆 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = (𝐴𝐹∪ 𝑆))
21	17, 20	syl 18	. . 3 ⊢ (𝑆 ∈ 𝑇 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = (𝐴𝐹∪ 𝑆))
22	21	3ad2ant1 1151	. 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 4972	. . 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 2803	1 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450   ⊆ wss 3896  ∅c0 4277  ∪ cuni 4866  ∪ ciun 4950   ↦ cmpt 5185  Oncon0 6365  Lim wlim 6366  ‘cfv 6543  (class class class)co 7424

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-ord 6368  df-on 6369  df-lim 6370  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427

This theorem is used by:  onoviun  8358