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Theorem onovuni 8382
Description: A variant of onfununi 8381 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
onovuni.1 (Lim 𝑦 → (𝐴𝐹𝑦) = 𝑥𝑦 (𝐴𝐹𝑥))
onovuni.2 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦))
Assertion
Ref Expression
onovuni ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹 𝑆) = 𝑥𝑆 (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.
StepHypRef Expression
1 onovuni.1 . . . 4 (Lim 𝑦 → (𝐴𝐹𝑦) = 𝑥𝑦 (𝐴𝐹𝑥))
2 oveq2 7439 . . . . . 6 (𝑧 = 𝑦 → (𝐴𝐹𝑧) = (𝐴𝐹𝑦))
3 eqid 2737 . . . . . 6 (𝑧 ∈ V ↦ (𝐴𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐴𝐹𝑧))
4 ovex 7464 . . . . . 6 (𝐴𝐹𝑦) ∈ V
52, 3, 4fvmpt 7016 . . . . 5 (𝑦 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦))
65elv 3485 . . . 4 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦)
7 oveq2 7439 . . . . . . . 8 (𝑧 = 𝑥 → (𝐴𝐹𝑧) = (𝐴𝐹𝑥))
8 ovex 7464 . . . . . . . 8 (𝐴𝐹𝑥) ∈ V
97, 3, 8fvmpt 7016 . . . . . . 7 (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
109elv 3485 . . . . . 6 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥)
1110a1i 11 . . . . 5 (𝑥𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
1211iuneq2i 5013 . . . 4 𝑥𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑦 (𝐴𝐹𝑥)
131, 6, 123eqtr4g 2802 . . 3 (Lim 𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = 𝑥𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
14 onovuni.2 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦))
1514, 10, 63sstr4g 4037 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) ⊆ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦))
1613, 15onfununi 8381 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
17 uniexg 7760 . . . 4 (𝑆𝑇 𝑆 ∈ V)
18 oveq2 7439 . . . . 5 (𝑧 = 𝑆 → (𝐴𝐹𝑧) = (𝐴𝐹 𝑆))
19 ovex 7464 . . . . 5 (𝐴𝐹 𝑆) ∈ V
2018, 3, 19fvmpt 7016 . . . 4 ( 𝑆 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
2117, 20syl 17 . . 3 (𝑆𝑇 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
22213ad2ant1 1134 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
2310a1i 11 . . . 4 (𝑥𝑆 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
2423iuneq2i 5013 . . 3 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑆 (𝐴𝐹𝑥)
2524a1i 11 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑆 (𝐴𝐹𝑥))
2616, 22, 253eqtr3d 2785 1 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹 𝑆) = 𝑥𝑆 (𝐴𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  wss 3951  c0 4333   cuni 4907   ciun 4991  cmpt 5225  Oncon0 6384  Lim wlim 6385  cfv 6561  (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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-lim 6389  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434
This theorem is referenced by:  onoviun  8383
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