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Theorem curfv 34866
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 6719 . . . . . . . . . 10 (𝐹 Fn (𝑉 × 𝑊) ↔ 𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
2 cureq 34862 . . . . . . . . . 10 (𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
31, 2sylbi 219 . . . . . . . . 9 (𝐹 Fn (𝑉 × 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
43adantr 483 . . . . . . . 8 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
5 fveq2 6665 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
65mpompt 7260 . . . . . . . . 9 (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) = (𝑥𝑉, 𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
7 fvex 6678 . . . . . . . . . . 11 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
87rgen2w 3151 . . . . . . . . . 10 𝑥𝑉𝑦𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
98a1i 11 . . . . . . . . 9 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → ∀𝑥𝑉𝑦𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V)
10 ne0i 4300 . . . . . . . . . 10 (𝐵𝑊𝑊 ≠ ∅)
1110adantl 484 . . . . . . . . 9 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → 𝑊 ≠ ∅)
126, 9, 11mpocurryd 7929 . . . . . . . 8 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
134, 12eqtrd 2856 . . . . . . 7 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry 𝐹 = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
14133adant2 1127 . . . . . 6 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → curry 𝐹 = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
1514fveq1d 6667 . . . . 5 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → (curry 𝐹𝐴) = ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
1615adantr 483 . . . 4 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → (curry 𝐹𝐴) = ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
17 mptexg 6978 . . . . . 6 (𝑊𝑋 → (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V)
18 opeq1 4797 . . . . . . . . 9 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1918fveq2d 6669 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹‘⟨𝑥, 𝑦⟩) = (𝐹‘⟨𝐴, 𝑦⟩))
2019mpteq2dv 5155 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
21 eqid 2821 . . . . . . 7 (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
2220, 21fvmptg 6761 . . . . . 6 ((𝐴𝑉 ∧ (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2317, 22sylan2 594 . . . . 5 ((𝐴𝑉𝑊𝑋) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
24233ad2antl2 1182 . . . 4 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2516, 24eqtrd 2856 . . 3 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → (curry 𝐹𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2625fveq1d 6667 . 2 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵))
27 opeq2 4798 . . . . . . 7 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2827fveq2d 6669 . . . . . 6 (𝑦 = 𝐵 → (𝐹‘⟨𝐴, 𝑦⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
29 eqid 2821 . . . . . 6 (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))
30 fvex 6678 . . . . . 6 (𝐹‘⟨𝐴, 𝐵⟩) ∈ V
3128, 29, 30fvmpt 6763 . . . . 5 (𝐵𝑊 → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐹‘⟨𝐴, 𝐵⟩))
32 df-ov 7153 . . . . 5 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
3331, 32syl6eqr 2874 . . . 4 (𝐵𝑊 → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
34333ad2ant3 1131 . . 3 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
3534adantr 483 . 2 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
3626, 35eqtrd 2856 1 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  wne 3016  wral 3138  Vcvv 3495  c0 4291  cop 4567  cmpt 5139   × cxp 5548   Fn wfn 6345  cfv 6350  (class class class)co 7150  curry ccur 7925
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-cur 7927
This theorem is referenced by:  unccur  34869  matunitlindflem1  34882  matunitlindflem2  34883
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