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Theorem curfv 33834
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 6434 . . . . . . . . . 10 (𝐹 Fn (𝑉 × 𝑊) ↔ 𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
2 cureq 33830 . . . . . . . . . 10 (𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
31, 2sylbi 208 . . . . . . . . 9 (𝐹 Fn (𝑉 × 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
43adantr 472 . . . . . . . 8 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
5 fveq2 6379 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
65mpt2mpt 6954 . . . . . . . . 9 (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) = (𝑥𝑉, 𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
7 fvex 6392 . . . . . . . . . . 11 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
87rgen2w 3072 . . . . . . . . . 10 𝑥𝑉𝑦𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
98a1i 11 . . . . . . . . 9 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → ∀𝑥𝑉𝑦𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V)
10 ne0i 4087 . . . . . . . . . 10 (𝐵𝑊𝑊 ≠ ∅)
1110adantl 473 . . . . . . . . 9 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → 𝑊 ≠ ∅)
126, 9, 11mpt2curryd 7602 . . . . . . . 8 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
134, 12eqtrd 2799 . . . . . . 7 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry 𝐹 = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
14133adant2 1161 . . . . . 6 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → curry 𝐹 = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
1514fveq1d 6381 . . . . 5 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → (curry 𝐹𝐴) = ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
1615adantr 472 . . . 4 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → (curry 𝐹𝐴) = ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
17 mptexg 6681 . . . . . 6 (𝑊𝑋 → (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V)
18 opeq1 4561 . . . . . . . . 9 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1918fveq2d 6383 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹‘⟨𝑥, 𝑦⟩) = (𝐹‘⟨𝐴, 𝑦⟩))
2019mpteq2dv 4906 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
21 eqid 2765 . . . . . . 7 (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
2220, 21fvmptg 6473 . . . . . 6 ((𝐴𝑉 ∧ (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2317, 22sylan2 586 . . . . 5 ((𝐴𝑉𝑊𝑋) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
24233ad2antl2 1237 . . . 4 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2516, 24eqtrd 2799 . . 3 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → (curry 𝐹𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2625fveq1d 6381 . 2 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵))
27 opeq2 4562 . . . . . . 7 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2827fveq2d 6383 . . . . . 6 (𝑦 = 𝐵 → (𝐹‘⟨𝐴, 𝑦⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
29 eqid 2765 . . . . . 6 (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))
30 fvex 6392 . . . . . 6 (𝐹‘⟨𝐴, 𝐵⟩) ∈ V
3128, 29, 30fvmpt 6475 . . . . 5 (𝐵𝑊 → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐹‘⟨𝐴, 𝐵⟩))
32 df-ov 6849 . . . . 5 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
3331, 32syl6eqr 2817 . . . 4 (𝐵𝑊 → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
34333ad2ant3 1165 . . 3 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
3534adantr 472 . 2 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
3626, 35eqtrd 2799 1 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  Vcvv 3350  c0 4081  cop 4342  cmpt 4890   × cxp 5277   Fn wfn 6065  cfv 6070  (class class class)co 6846  curry ccur 7598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-cur 7600
This theorem is referenced by:  unccur  33837  matunitlindflem1  33850  matunitlindflem2  33851
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