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Theorem curfv 36971
Description: Value of currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
Assertion
Ref Expression
curfv (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))

Proof of Theorem curfv
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffn5 6941 . . . . . . . . . 10 (𝐹 Fn (𝑉 × 𝑊) ↔ 𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
2 cureq 36967 . . . . . . . . . 10 (𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
31, 2sylbi 216 . . . . . . . . 9 (𝐹 Fn (𝑉 × 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
43adantr 480 . . . . . . . 8 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
5 fveq2 6882 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
65mpompt 7515 . . . . . . . . 9 (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) = (𝑥𝑉, 𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
7 fvex 6895 . . . . . . . . . . 11 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
87rgen2w 3058 . . . . . . . . . 10 𝑥𝑉𝑦𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
98a1i 11 . . . . . . . . 9 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → ∀𝑥𝑉𝑦𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V)
10 ne0i 4327 . . . . . . . . . 10 (𝐵𝑊𝑊 ≠ ∅)
1110adantl 481 . . . . . . . . 9 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → 𝑊 ≠ ∅)
126, 9, 11mpocurryd 8250 . . . . . . . 8 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
134, 12eqtrd 2764 . . . . . . 7 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry 𝐹 = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
14133adant2 1128 . . . . . 6 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → curry 𝐹 = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
1514fveq1d 6884 . . . . 5 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → (curry 𝐹𝐴) = ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
1615adantr 480 . . . 4 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → (curry 𝐹𝐴) = ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
17 mptexg 7215 . . . . . 6 (𝑊𝑋 → (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V)
18 opeq1 4866 . . . . . . . . 9 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1918fveq2d 6886 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹‘⟨𝑥, 𝑦⟩) = (𝐹‘⟨𝐴, 𝑦⟩))
2019mpteq2dv 5241 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
21 eqid 2724 . . . . . . 7 (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
2220, 21fvmptg 6987 . . . . . 6 ((𝐴𝑉 ∧ (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2317, 22sylan2 592 . . . . 5 ((𝐴𝑉𝑊𝑋) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
24233ad2antl2 1183 . . . 4 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2516, 24eqtrd 2764 . . 3 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → (curry 𝐹𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2625fveq1d 6884 . 2 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵))
27 opeq2 4867 . . . . . . 7 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2827fveq2d 6886 . . . . . 6 (𝑦 = 𝐵 → (𝐹‘⟨𝐴, 𝑦⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
29 eqid 2724 . . . . . 6 (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))
30 fvex 6895 . . . . . 6 (𝐹‘⟨𝐴, 𝐵⟩) ∈ V
3128, 29, 30fvmpt 6989 . . . . 5 (𝐵𝑊 → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐹‘⟨𝐴, 𝐵⟩))
32 df-ov 7405 . . . . 5 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
3331, 32eqtr4di 2782 . . . 4 (𝐵𝑊 → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
34333ad2ant3 1132 . . 3 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
3534adantr 480 . 2 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
3626, 35eqtrd 2764 1 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2932  wral 3053  Vcvv 3466  c0 4315  cop 4627  cmpt 5222   × cxp 5665   Fn wfn 6529  cfv 6534  (class class class)co 7402  curry ccur 8246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-cur 8248
This theorem is referenced by:  unccur  36974  matunitlindflem1  36987  matunitlindflem2  36988
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