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Theorem curfv 37073
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 6957 . . . . . . . . . 10 (𝐹 Fn (𝑉 × 𝑊) ↔ 𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
2 cureq 37069 . . . . . . . . . 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 6897 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
65mpompt 7534 . . . . . . . . 9 (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) = (𝑥𝑉, 𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
7 fvex 6910 . . . . . . . . . . 11 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
87rgen2w 3063 . . . . . . . . . 10 𝑥𝑉𝑦𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
98a1i 11 . . . . . . . . 9 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → ∀𝑥𝑉𝑦𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V)
10 ne0i 4335 . . . . . . . . . 10 (𝐵𝑊𝑊 ≠ ∅)
1110adantl 481 . . . . . . . . 9 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → 𝑊 ≠ ∅)
126, 9, 11mpocurryd 8274 . . . . . . . 8 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
134, 12eqtrd 2768 . . . . . . 7 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry 𝐹 = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
14133adant2 1129 . . . . . 6 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → curry 𝐹 = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
1514fveq1d 6899 . . . . 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 7233 . . . . . 6 (𝑊𝑋 → (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V)
18 opeq1 4874 . . . . . . . . 9 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1918fveq2d 6901 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹‘⟨𝑥, 𝑦⟩) = (𝐹‘⟨𝐴, 𝑦⟩))
2019mpteq2dv 5250 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
21 eqid 2728 . . . . . . 7 (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
2220, 21fvmptg 7003 . . . . . 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 1184 . . . 4 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2516, 24eqtrd 2768 . . 3 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → (curry 𝐹𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2625fveq1d 6899 . 2 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵))
27 opeq2 4875 . . . . . . 7 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2827fveq2d 6901 . . . . . 6 (𝑦 = 𝐵 → (𝐹‘⟨𝐴, 𝑦⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
29 eqid 2728 . . . . . 6 (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))
30 fvex 6910 . . . . . 6 (𝐹‘⟨𝐴, 𝐵⟩) ∈ V
3128, 29, 30fvmpt 7005 . . . . 5 (𝐵𝑊 → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐹‘⟨𝐴, 𝐵⟩))
32 df-ov 7423 . . . . 5 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
3331, 32eqtr4di 2786 . . . 4 (𝐵𝑊 → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
34333ad2ant3 1133 . . 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 2768 1 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2937  wral 3058  Vcvv 3471  c0 4323  cop 4635  cmpt 5231   × cxp 5676   Fn wfn 6543  cfv 6548  (class class class)co 7420  curry ccur 8270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-cur 8272
This theorem is referenced by:  unccur  37076  matunitlindflem1  37089  matunitlindflem2  37090
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