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Theorem curfv 33721
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 6405 . . . . . . . . . 10 (𝐹 Fn (𝑉 × 𝑊) ↔ 𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
2 cureq 33717 . . . . . . . . . 10 (𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)))
31, 2sylbi 207 . . . . . . . . 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 6354 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
65mpt2mpt 6919 . . . . . . . . 9 (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) = (𝑥𝑉, 𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
7 fvex 6364 . . . . . . . . . . 11 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
87rgen2w 3064 . . . . . . . . . 10 𝑥𝑉𝑦𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
98a1i 11 . . . . . . . . 9 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → ∀𝑥𝑉𝑦𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V)
10 ne0i 4065 . . . . . . . . . 10 (𝐵𝑊𝑊 ≠ ∅)
1110adantl 473 . . . . . . . . 9 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → 𝑊 ≠ ∅)
126, 9, 11mpt2curryd 7566 . . . . . . . 8 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹𝑧)) = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
134, 12eqtrd 2795 . . . . . . 7 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵𝑊) → curry 𝐹 = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
14133adant2 1126 . . . . . 6 ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) → curry 𝐹 = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
1514fveq1d 6356 . . . . 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 6650 . . . . . 6 (𝑊𝑋 → (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V)
18 opeq1 4554 . . . . . . . . 9 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1918fveq2d 6358 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹‘⟨𝑥, 𝑦⟩) = (𝐹‘⟨𝐴, 𝑦⟩))
2019mpteq2dv 4898 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
21 eqid 2761 . . . . . . 7 (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
2220, 21fvmptg 6444 . . . . . 6 ((𝐴𝑉 ∧ (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2317, 22sylan2 492 . . . . 5 ((𝐴𝑉𝑊𝑋) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
24233ad2antl2 1202 . . . 4 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((𝑥𝑉 ↦ (𝑦𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2516, 24eqtrd 2795 . . 3 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → (curry 𝐹𝐴) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
2625fveq1d 6356 . 2 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵))
27 opeq2 4555 . . . . . . 7 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2827fveq2d 6358 . . . . . 6 (𝑦 = 𝐵 → (𝐹‘⟨𝐴, 𝑦⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
29 eqid 2761 . . . . . 6 (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) = (𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))
30 fvex 6364 . . . . . 6 (𝐹‘⟨𝐴, 𝐵⟩) ∈ V
3128, 29, 30fvmpt 6446 . . . . 5 (𝐵𝑊 → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐹‘⟨𝐴, 𝐵⟩))
32 df-ov 6818 . . . . 5 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
3331, 32syl6eqr 2813 . . . 4 (𝐵𝑊 → ((𝑦𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
34333ad2ant3 1130 . . 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 2795 1 (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴𝑉𝐵𝑊) ∧ 𝑊𝑋) → ((curry 𝐹𝐴)‘𝐵) = (𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2140  wne 2933  wral 3051  Vcvv 3341  c0 4059  cop 4328  cmpt 4882   × cxp 5265   Fn wfn 6045  cfv 6050  (class class class)co 6815  curry ccur 7562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-cur 7564
This theorem is referenced by:  unccur  33724  matunitlindflem1  33737  matunitlindflem2  33738
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