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Theorem onovuni 7980
 Description: A variant of onfununi 7979 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 7153 . . . . . 6 (𝑧 = 𝑦 → (𝐴𝐹𝑧) = (𝐴𝐹𝑦))
3 eqid 2798 . . . . . 6 (𝑧 ∈ V ↦ (𝐴𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐴𝐹𝑧))
4 ovex 7178 . . . . . 6 (𝐴𝐹𝑦) ∈ V
52, 3, 4fvmpt 6755 . . . . 5 (𝑦 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦))
65elv 3447 . . . 4 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦)
7 oveq2 7153 . . . . . . . 8 (𝑧 = 𝑥 → (𝐴𝐹𝑧) = (𝐴𝐹𝑥))
8 ovex 7178 . . . . . . . 8 (𝐴𝐹𝑥) ∈ V
97, 3, 8fvmpt 6755 . . . . . . 7 (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
109elv 3447 . . . . . 6 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥)
1110a1i 11 . . . . 5 (𝑥𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
1211iuneq2i 4906 . . . 4 𝑥𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑦 (𝐴𝐹𝑥)
131, 6, 123eqtr4g 2858 . . 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 3962 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) ⊆ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦))
1613, 15onfununi 7979 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
17 uniexg 7459 . . . 4 (𝑆𝑇 𝑆 ∈ V)
18 oveq2 7153 . . . . 5 (𝑧 = 𝑆 → (𝐴𝐹𝑧) = (𝐴𝐹 𝑆))
19 ovex 7178 . . . . 5 (𝐴𝐹 𝑆) ∈ V
2018, 3, 19fvmpt 6755 . . . 4 ( 𝑆 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
2117, 20syl 17 . . 3 (𝑆𝑇 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
22213ad2ant1 1130 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
2310a1i 11 . . . 4 (𝑥𝑆 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
2423iuneq2i 4906 . . 3 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑆 (𝐴𝐹𝑥)
2524a1i 11 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑆 (𝐴𝐹𝑥))
2616, 22, 253eqtr3d 2841 1 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹 𝑆) = 𝑥𝑆 (𝐴𝐹𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3442   ⊆ wss 3883  ∅c0 4246  ∪ cuni 4804  ∪ ciun 4885   ↦ cmpt 5114  Oncon0 6166  Lim wlim 6167  ‘cfv 6332  (class class class)co 7145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-ord 6169  df-on 6170  df-lim 6171  df-iota 6291  df-fun 6334  df-fv 6340  df-ov 7148 This theorem is referenced by:  onoviun  7981
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