Theorem onovuni 8268

Description: A variant of onfununi 8267 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 7360	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧) = (𝐴𝐹𝑦))
3		eqid 2733	. . . . . 6 ⊢ (𝑧 ∈ V ↦ (𝐴𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐴𝐹𝑧))
4		ovex 7385	. . . . . 6 ⊢ (𝐴𝐹𝑦) ∈ V
5	2, 3, 4	fvmpt 6935	. . . . 5 ⊢ (𝑦 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦))
6	5	elv 3442	. . . 4 ⊢ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦)
7		oveq2 7360	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝐴𝐹𝑧) = (𝐴𝐹𝑥))
8		ovex 7385	. . . . . . . 8 ⊢ (𝐴𝐹𝑥) ∈ V
9	7, 3, 8	fvmpt 6935	. . . . . . 7 ⊢ (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
10	9	elv 3442	. . . . . 6 ⊢ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥)
11	10	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
12	11	iuneq2i 4963	. . . 4 ⊢ ∪ 𝑥 ∈ 𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = ∪ 𝑥 ∈ 𝑦 (𝐴𝐹𝑥)
13	1, 6, 12	3eqtr4g 2793	. . 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 3984	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) ⊆ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦))
16	13, 15	onfununi 8267	. 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
17		uniexg 7679	. . . 4 ⊢ (𝑆 ∈ 𝑇 → ∪ 𝑆 ∈ V)
18		oveq2 7360	. . . . 5 ⊢ (𝑧 = ∪ 𝑆 → (𝐴𝐹𝑧) = (𝐴𝐹∪ 𝑆))
19		ovex 7385	. . . . 5 ⊢ (𝐴𝐹∪ 𝑆) ∈ V
20	18, 3, 19	fvmpt 6935	. . . 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 1133	. 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 4963	. . 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 2776	1 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  Vcvv 3437   ⊆ wss 3898  ∅c0 4282  ∪ cuni 4858  ∪ ciun 4941   ↦ cmpt 5174  Oncon0 6311  Lim wlim 6312  ‘cfv 6486  (class class class)co 7352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-ord 6314  df-on 6315  df-lim 6316  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355

This theorem is referenced by:  onoviun  8269