Theorem onovuni 8325

Description: A variant of onfununi 8324 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 7418	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧) = (𝐴𝐹𝑦))
3		eqid 2763	. . . . . 6 ⊢ (𝑧 ∈ V ↦ (𝐴𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐴𝐹𝑧))
4		ovex 7443	. . . . . 6 ⊢ (𝐴𝐹𝑦) ∈ V
5	2, 3, 4	fvmpt 6989	. . . . 5 ⊢ (𝑦 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦))
6	5	elv 3460	. . . 4 ⊢ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦)
7		oveq2 7418	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝐴𝐹𝑧) = (𝐴𝐹𝑥))
8		ovex 7443	. . . . . . . 8 ⊢ (𝐴𝐹𝑥) ∈ V
9	7, 3, 8	fvmpt 6989	. . . . . . 7 ⊢ (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
10	9	elv 3460	. . . . . 6 ⊢ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥)
11	10	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
12	11	iuneq2i 4978	. . . 4 ⊢ ∪ 𝑥 ∈ 𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = ∪ 𝑥 ∈ 𝑦 (𝐴𝐹𝑥)
13	1, 6, 12	3eqtr4g 2823	. . 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 3990	. . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) ⊆ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦))
16	13, 15	onfununi 8324	. 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
17		uniexg 7738	. . . 4 ⊢ (𝑆 ∈ 𝑇 → ∪ 𝑆 ∈ V)
18		oveq2 7418	. . . . 5 ⊢ (𝑧 = ∪ 𝑆 → (𝐴𝐹𝑧) = (𝐴𝐹∪ 𝑆))
19		ovex 7443	. . . . 5 ⊢ (𝐴𝐹∪ 𝑆) ∈ V
20	18, 3, 19	fvmpt 6989	. . . 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 4978	. . 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 2806	1 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥))

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  ∪ cuni 4872  ∪ ciun 4956   ↦ cmpt 5192  Oncon0 6360  Lim wlim 6361  ‘cfv 6536  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-ord 6363  df-on 6364  df-lim 6365  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413

This theorem is referenced by:  onoviun  8326