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Theorem mpocurryd 7786
Description: The currying of an operation given in maps-to notation, splitting the operation (function of two arguments) into a function of the first argument, producing a function over the second argument. (Contributed by AV, 27-Oct-2019.)
Hypotheses
Ref Expression
mpocurryd.f 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
mpocurryd.c (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
mpocurryd.n (𝜑𝑌 ≠ ∅)
Assertion
Ref Expression
mpocurryd (𝜑 → curry 𝐹 = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem mpocurryd
Dummy variables 𝑎 𝑏 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cur 7784 . 2 curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧})
2 mpocurryd.c . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐶𝑉)
3 mpocurryd.f . . . . . . . 8 𝐹 = (𝑥𝑋, 𝑦𝑌𝐶)
43dmmpoga 7627 . . . . . . 7 (∀𝑥𝑋𝑦𝑌 𝐶𝑉 → dom 𝐹 = (𝑋 × 𝑌))
52, 4syl 17 . . . . . 6 (𝜑 → dom 𝐹 = (𝑋 × 𝑌))
65dmeqd 5660 . . . . 5 (𝜑 → dom dom 𝐹 = dom (𝑋 × 𝑌))
7 mpocurryd.n . . . . . 6 (𝜑𝑌 ≠ ∅)
8 dmxp 5681 . . . . . 6 (𝑌 ≠ ∅ → dom (𝑋 × 𝑌) = 𝑋)
97, 8syl 17 . . . . 5 (𝜑 → dom (𝑋 × 𝑌) = 𝑋)
106, 9eqtrd 2831 . . . 4 (𝜑 → dom dom 𝐹 = 𝑋)
1110mpteq1d 5049 . . 3 (𝜑 → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝑋 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}))
12 df-mpt 5042 . . . . 5 (𝑦𝑌𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝐶)}
133mpofun 7132 . . . . . . . 8 Fun 𝐹
14 funbrfv2b 6591 . . . . . . . 8 (Fun 𝐹 → (⟨𝑥, 𝑦𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
1513, 14mp1i 13 . . . . . . 7 ((𝜑𝑥𝑋) → (⟨𝑥, 𝑦𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
165adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝑋) → dom 𝐹 = (𝑋 × 𝑌))
1716eleq2d 2868 . . . . . . . . 9 ((𝜑𝑥𝑋) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌)))
18 opelxp 5479 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌) ↔ (𝑥𝑋𝑦𝑌))
1917, 18syl6bb 288 . . . . . . . 8 ((𝜑𝑥𝑋) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ (𝑥𝑋𝑦𝑌)))
2019anbi1d 629 . . . . . . 7 ((𝜑𝑥𝑋) → ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ ((𝑥𝑋𝑦𝑌) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
21 an21 640 . . . . . . . 8 (((𝑥𝑋𝑦𝑌) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦𝑌 ∧ (𝑥𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
22 ibar 529 . . . . . . . . . . . . 13 (𝑥𝑋 → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ (𝑥𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
2322bicomd 224 . . . . . . . . . . . 12 (𝑥𝑋 → ((𝑥𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
2423adantl 482 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → ((𝑥𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
2524adantr 481 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝑥𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
26 df-ov 7019 . . . . . . . . . . . . 13 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
27 nfcv 2949 . . . . . . . . . . . . . . . . 17 𝑎𝐶
28 nfcv 2949 . . . . . . . . . . . . . . . . 17 𝑏𝐶
29 nfcv 2949 . . . . . . . . . . . . . . . . . 18 𝑥𝑏
30 nfcsb1v 3833 . . . . . . . . . . . . . . . . . 18 𝑥𝑎 / 𝑥𝐶
3129, 30nfcsb 3835 . . . . . . . . . . . . . . . . 17 𝑥𝑏 / 𝑦𝑎 / 𝑥𝐶
32 nfcsb1v 3833 . . . . . . . . . . . . . . . . 17 𝑦𝑏 / 𝑦𝑎 / 𝑥𝐶
33 csbeq1a 3824 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
34 csbeq1a 3824 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏𝑎 / 𝑥𝐶 = 𝑏 / 𝑦𝑎 / 𝑥𝐶)
3533, 34sylan9eq 2851 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐶 = 𝑏 / 𝑦𝑎 / 𝑥𝐶)
3627, 28, 31, 32, 35cbvmpo 7104 . . . . . . . . . . . . . . . 16 (𝑥𝑋, 𝑦𝑌𝐶) = (𝑎𝑋, 𝑏𝑌𝑏 / 𝑦𝑎 / 𝑥𝐶)
373, 36eqtri 2819 . . . . . . . . . . . . . . 15 𝐹 = (𝑎𝑋, 𝑏𝑌𝑏 / 𝑦𝑎 / 𝑥𝐶)
3837a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐹 = (𝑎𝑋, 𝑏𝑌𝑏 / 𝑦𝑎 / 𝑥𝐶))
3933eqcomd 2801 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎𝑎 / 𝑥𝐶 = 𝐶)
4039equcoms 2004 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥𝑎 / 𝑥𝐶 = 𝐶)
4140csbeq2dv 3818 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝑏 / 𝑦𝐶)
42 csbeq1a 3824 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑏𝐶 = 𝑏 / 𝑦𝐶)
4342eqcomd 2801 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝑏 / 𝑦𝐶 = 𝐶)
4443equcoms 2004 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑦𝑏 / 𝑦𝐶 = 𝐶)
4541, 44sylan9eq 2851 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑥𝑏 = 𝑦) → 𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐶)
4645adantl 482 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑋) ∧ 𝑦𝑌) ∧ (𝑎 = 𝑥𝑏 = 𝑦)) → 𝑏 / 𝑦𝑎 / 𝑥𝐶 = 𝐶)
47 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → 𝑥𝑋)
4847adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑥𝑋)
49 simpr 485 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
50 rsp2 3180 . . . . . . . . . . . . . . . 16 (∀𝑥𝑋𝑦𝑌 𝐶𝑉 → ((𝑥𝑋𝑦𝑌) → 𝐶𝑉))
512, 50syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐶𝑉))
5251impl 456 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐶𝑉)
5338, 46, 48, 49, 52ovmpod 7158 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝑥𝐹𝑦) = 𝐶)
5426, 53syl5eqr 2845 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
5554eqeq1d 2797 . . . . . . . . . . 11 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧𝐶 = 𝑧))
56 eqcom 2802 . . . . . . . . . . 11 (𝐶 = 𝑧𝑧 = 𝐶)
5755, 56syl6bb 288 . . . . . . . . . 10 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧𝑧 = 𝐶))
5825, 57bitrd 280 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → ((𝑥𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ 𝑧 = 𝐶))
5958pm5.32da 579 . . . . . . . 8 ((𝜑𝑥𝑋) → ((𝑦𝑌 ∧ (𝑥𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)) ↔ (𝑦𝑌𝑧 = 𝐶)))
6021, 59syl5bb 284 . . . . . . 7 ((𝜑𝑥𝑋) → (((𝑥𝑋𝑦𝑌) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦𝑌𝑧 = 𝐶)))
6115, 20, 603bitrrd 307 . . . . . 6 ((𝜑𝑥𝑋) → ((𝑦𝑌𝑧 = 𝐶) ↔ ⟨𝑥, 𝑦𝐹𝑧))
6261opabbidv 5028 . . . . 5 ((𝜑𝑥𝑋) → {⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝐶)} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧})
6312, 62syl5req 2844 . . . 4 ((𝜑𝑥𝑋) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧} = (𝑦𝑌𝐶))
6463mpteq2dva 5055 . . 3 (𝜑 → (𝑥𝑋 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
6511, 64eqtrd 2831 . 2 (𝜑 → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
661, 65syl5eq 2843 1 (𝜑 → curry 𝐹 = (𝑥𝑋 ↦ (𝑦𝑌𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wne 2984  wral 3105  csb 3811  c0 4211  cop 4478   class class class wbr 4962  {copab 5024  cmpt 5041   × cxp 5441  dom cdm 5443  Fun wfun 6219  cfv 6225  (class class class)co 7016  cmpo 7018  curry ccur 7782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-cur 7784
This theorem is referenced by:  mpocurryvald  7787  curfv  34422
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