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Theorem onovuni 7326
 Description: A variant of onfununi 7325 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 vex 3176 . . . . 5 𝑦 ∈ V
3 oveq2 6557 . . . . . 6 (𝑧 = 𝑦 → (𝐴𝐹𝑧) = (𝐴𝐹𝑦))
4 eqid 2610 . . . . . 6 (𝑧 ∈ V ↦ (𝐴𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐴𝐹𝑧))
5 ovex 6577 . . . . . 6 (𝐴𝐹𝑦) ∈ V
63, 4, 5fvmpt 6191 . . . . 5 (𝑦 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦))
72, 6ax-mp 5 . . . 4 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦)
8 vex 3176 . . . . . . 7 𝑥 ∈ V
9 oveq2 6557 . . . . . . . 8 (𝑧 = 𝑥 → (𝐴𝐹𝑧) = (𝐴𝐹𝑥))
10 ovex 6577 . . . . . . . 8 (𝐴𝐹𝑥) ∈ V
119, 4, 10fvmpt 6191 . . . . . . 7 (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
128, 11ax-mp 5 . . . . . 6 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥)
1312a1i 11 . . . . 5 (𝑥𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
1413iuneq2i 4475 . . . 4 𝑥𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑦 (𝐴𝐹𝑥)
151, 7, 143eqtr4g 2669 . . 3 (Lim 𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = 𝑥𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
16 onovuni.2 . . . 4 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦))
1716, 12, 73sstr4g 3609 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) ⊆ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦))
1815, 17onfununi 7325 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
19 uniexg 6853 . . . 4 (𝑆𝑇 𝑆 ∈ V)
20 oveq2 6557 . . . . 5 (𝑧 = 𝑆 → (𝐴𝐹𝑧) = (𝐴𝐹 𝑆))
21 ovex 6577 . . . . 5 (𝐴𝐹 𝑆) ∈ V
2220, 4, 21fvmpt 6191 . . . 4 ( 𝑆 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
2319, 22syl 17 . . 3 (𝑆𝑇 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
24233ad2ant1 1075 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
2512a1i 11 . . . 4 (𝑥𝑆 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
2625iuneq2i 4475 . . 3 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑆 (𝐴𝐹𝑥)
2726a1i 11 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑆 (𝐴𝐹𝑥))
2818, 24, 273eqtr3d 2652 1 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹 𝑆) = 𝑥𝑆 (𝐴𝐹𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643  Oncon0 5640  Lim wlim 5641  ‘cfv 5804  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-ord 5643  df-on 5644  df-lim 5645  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552 This theorem is referenced by:  onoviun  7327
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