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Theorem onovuni 7298
Description: A variant of onfununi 7297 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 3170 . . . . 5 𝑦 ∈ V
3 oveq2 6530 . . . . . 6 (𝑧 = 𝑦 → (𝐴𝐹𝑧) = (𝐴𝐹𝑦))
4 eqid 2604 . . . . . 6 (𝑧 ∈ V ↦ (𝐴𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐴𝐹𝑧))
5 ovex 6550 . . . . . 6 (𝐴𝐹𝑦) ∈ V
63, 4, 5fvmpt 6171 . . . . 5 (𝑦 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦))
72, 6ax-mp 5 . . . 4 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦)
8 vex 3170 . . . . . . 7 𝑥 ∈ V
9 oveq2 6530 . . . . . . . 8 (𝑧 = 𝑥 → (𝐴𝐹𝑧) = (𝐴𝐹𝑥))
10 ovex 6550 . . . . . . . 8 (𝐴𝐹𝑥) ∈ V
119, 4, 10fvmpt 6171 . . . . . . 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 4464 . . . 4 𝑥𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑦 (𝐴𝐹𝑥)
151, 7, 143eqtr4g 2663 . . 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 3603 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) ⊆ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦))
1815, 17onfununi 7297 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥))
19 uniexg 6825 . . . 4 (𝑆𝑇 𝑆 ∈ V)
20 oveq2 6530 . . . . 5 (𝑧 = 𝑆 → (𝐴𝐹𝑧) = (𝐴𝐹 𝑆))
21 ovex 6550 . . . . 5 (𝐴𝐹 𝑆) ∈ V
2220, 4, 21fvmpt 6171 . . . 4 ( 𝑆 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
2319, 22syl 17 . . 3 (𝑆𝑇 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
24233ad2ant1 1074 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘ 𝑆) = (𝐴𝐹 𝑆))
2512a1i 11 . . . 4 (𝑥𝑆 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥))
2625iuneq2i 4464 . . 3 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑆 (𝐴𝐹𝑥)
2726a1i 11 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑥𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = 𝑥𝑆 (𝐴𝐹𝑥))
2818, 24, 273eqtr3d 2646 1 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹 𝑆) = 𝑥𝑆 (𝐴𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  wcel 1975  wne 2774  Vcvv 3167  wss 3534  c0 3868   cuni 4361   ciun 4444  cmpt 4632  Oncon0 5621  Lim wlim 5622  cfv 5785  (class class class)co 6522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-ord 5624  df-on 5625  df-lim 5626  df-iota 5749  df-fun 5787  df-fv 5793  df-ov 6525
This theorem is referenced by:  onoviun  7299
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