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Theorem fnotovbOLD 6841
 Description: Old proof of fnotovb 6840 obsolete as of 15-Feb-2022. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fnotovbOLD ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))

Proof of Theorem fnotovbOLD
StepHypRef Expression
1 opelxpi 5288 . . . 4 ((𝐶𝐴𝐷𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵))
2 fnopfvb 6378 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → ((𝐹‘⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
31, 2sylan2 580 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶𝐴𝐷𝐵)) → ((𝐹‘⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
433impb 1107 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ((𝐹‘⟨𝐶, 𝐷⟩) = 𝑅 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹))
5 df-ov 6796 . . 3 (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
65eqeq1i 2776 . 2 ((𝐶𝐹𝐷) = 𝑅 ↔ (𝐹‘⟨𝐶, 𝐷⟩) = 𝑅)
7 df-ot 4325 . . 3 𝐶, 𝐷, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑅
87eleq1i 2841 . 2 (⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹 ↔ ⟨⟨𝐶, 𝐷⟩, 𝑅⟩ ∈ 𝐹)
94, 6, 83bitr4g 303 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ ⟨𝐶, 𝐷, 𝑅⟩ ∈ 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ⟨cop 4322  ⟨cotp 4324   × cxp 5247   Fn wfn 6026  ‘cfv 6031  (class class class)co 6793 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-ot 4325  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ov 6796 This theorem is referenced by: (None)
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