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Theorem ovg 7598
Description: The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
ovg.1 (𝑥 = 𝐴 → (𝜑𝜓))
ovg.2 (𝑦 = 𝐵 → (𝜓𝜒))
ovg.3 (𝑧 = 𝐶 → (𝜒𝜃))
ovg.4 ((𝜏 ∧ (𝑥𝑅𝑦𝑆)) → ∃!𝑧𝜑)
ovg.5 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
Assertion
Ref Expression
ovg ((𝜏 ∧ (𝐴𝑅𝐵𝑆𝐶𝐷)) → ((𝐴𝐹𝐵) = 𝐶𝜃))
Distinct variable groups:   𝜓,𝑥   𝜒,𝑥,𝑦   𝜃,𝑥,𝑦,𝑧   𝜏,𝑥,𝑦   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑦,𝑧)   𝜒(𝑧)   𝜏(𝑧)   𝐷(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem ovg
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-ov 7434 . . . . 5 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 ovg.5 . . . . . 6 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
32fveq1i 6908 . . . . 5 (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
41, 3eqtri 2763 . . . 4 (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
54eqeq1i 2740 . . 3 ((𝐴𝐹𝐵) = 𝐶 ↔ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶)
6 eqeq2 2747 . . . . . . . . . 10 (𝑐 = 𝐶 → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝑐 ↔ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶))
7 opeq2 4879 . . . . . . . . . . 11 (𝑐 = 𝐶 → ⟨⟨𝐴, 𝐵⟩, 𝑐⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
87eleq1d 2824 . . . . . . . . . 10 (𝑐 = 𝐶 → (⟨⟨𝐴, 𝐵⟩, 𝑐⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}))
96, 8bibi12d 345 . . . . . . . . 9 (𝑐 = 𝐶 → ((({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝑐 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑐⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}) ↔ (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)})))
109imbi2d 340 . . . . . . . 8 (𝑐 = 𝐶 → (((𝜏 ∧ (𝐴𝑅𝐵𝑆)) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝑐 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑐⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)})) ↔ ((𝜏 ∧ (𝐴𝑅𝐵𝑆)) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}))))
11 ovg.4 . . . . . . . . . . . 12 ((𝜏 ∧ (𝑥𝑅𝑦𝑆)) → ∃!𝑧𝜑)
1211ex 412 . . . . . . . . . . 11 (𝜏 → ((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑))
1312alrimivv 1926 . . . . . . . . . 10 (𝜏 → ∀𝑥𝑦((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑))
14 fnoprabg 7556 . . . . . . . . . 10 (∀𝑥𝑦((𝑥𝑅𝑦𝑆) → ∃!𝑧𝜑) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
1513, 14syl 17 . . . . . . . . 9 (𝜏 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
16 eleq1 2827 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
1716anbi1d 631 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝑦𝑆)))
18 eleq1 2827 . . . . . . . . . . . 12 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
1918anbi2d 630 . . . . . . . . . . 11 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
2017, 19opelopabg 5548 . . . . . . . . . 10 ((𝐴𝑅𝐵𝑆) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ↔ (𝐴𝑅𝐵𝑆)))
2120ibir 268 . . . . . . . . 9 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)})
22 fnopfvb 6961 . . . . . . . . 9 (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)} ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑅𝑦𝑆)}) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝑐 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑐⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}))
2315, 21, 22syl2an 596 . . . . . . . 8 ((𝜏 ∧ (𝐴𝑅𝐵𝑆)) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝑐 ↔ ⟨⟨𝐴, 𝐵⟩, 𝑐⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}))
2410, 23vtoclg 3554 . . . . . . 7 (𝐶𝐷 → ((𝜏 ∧ (𝐴𝑅𝐵𝑆)) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)})))
2524com12 32 . . . . . 6 ((𝜏 ∧ (𝐴𝑅𝐵𝑆)) → (𝐶𝐷 → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)})))
2625exp32 420 . . . . 5 (𝜏 → (𝐴𝑅 → (𝐵𝑆 → (𝐶𝐷 → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)})))))
27263imp2 1348 . . . 4 ((𝜏 ∧ (𝐴𝑅𝐵𝑆𝐶𝐷)) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}))
28 ovg.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
2917, 28anbi12d 632 . . . . . 6 (𝑥 = 𝐴 → (((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝐴𝑅𝑦𝑆) ∧ 𝜓)))
30 ovg.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
3119, 30anbi12d 632 . . . . . 6 (𝑦 = 𝐵 → (((𝐴𝑅𝑦𝑆) ∧ 𝜓) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜒)))
32 ovg.3 . . . . . . 7 (𝑧 = 𝐶 → (𝜒𝜃))
3332anbi2d 630 . . . . . 6 (𝑧 = 𝐶 → (((𝐴𝑅𝐵𝑆) ∧ 𝜒) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
3429, 31, 33eloprabg 7543 . . . . 5 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
3534adantl 481 . . . 4 ((𝜏 ∧ (𝐴𝑅𝐵𝑆𝐶𝐷)) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)} ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
3627, 35bitrd 279 . . 3 ((𝜏 ∧ (𝐴𝑅𝐵𝑆𝐶𝐷)) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
375, 36bitrid 283 . 2 ((𝜏 ∧ (𝐴𝑅𝐵𝑆𝐶𝐷)) → ((𝐴𝐹𝐵) = 𝐶 ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
38 biidd 262 . . . . 5 ((𝐴𝑅𝐵𝑆) → (((𝐴𝑅𝐵𝑆) ∧ 𝜃) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜃)))
3938bianabs 541 . . . 4 ((𝐴𝑅𝐵𝑆) → (((𝐴𝑅𝐵𝑆) ∧ 𝜃) ↔ 𝜃))
40393adant3 1131 . . 3 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (((𝐴𝑅𝐵𝑆) ∧ 𝜃) ↔ 𝜃))
4140adantl 481 . 2 ((𝜏 ∧ (𝐴𝑅𝐵𝑆𝐶𝐷)) → (((𝐴𝑅𝐵𝑆) ∧ 𝜃) ↔ 𝜃))
4237, 41bitrd 279 1 ((𝜏 ∧ (𝐴𝑅𝐵𝑆𝐶𝐷)) → ((𝐴𝐹𝐵) = 𝐶𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  ∃!weu 2566  cop 4637  {copab 5210   Fn wfn 6558  cfv 6563  (class class class)co 7431  {coprab 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-oprab 7435
This theorem is referenced by: (None)
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