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Theorem curry2 7232
 Description: Composition with ◡(1st ↾ (V × {𝐶})) turns any binary operation 𝐹 with a constant second operand into a function 𝐺 of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)
Hypothesis
Ref Expression
curry2.1 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
Assertion
Ref Expression
curry2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝐺

Proof of Theorem curry2
StepHypRef Expression
1 fnfun 5956 . . . . 5 (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹)
2 1stconst 7225 . . . . . 6 (𝐶𝐵 → (1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V)
3 dff1o3 6110 . . . . . . 7 ((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V ↔ ((1st ↾ (V × {𝐶})):(V × {𝐶})–onto→V ∧ Fun (1st ↾ (V × {𝐶}))))
43simprbi 480 . . . . . 6 ((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V → Fun (1st ↾ (V × {𝐶})))
52, 4syl 17 . . . . 5 (𝐶𝐵 → Fun (1st ↾ (V × {𝐶})))
6 funco 5896 . . . . 5 ((Fun 𝐹 ∧ Fun (1st ↾ (V × {𝐶}))) → Fun (𝐹(1st ↾ (V × {𝐶}))))
71, 5, 6syl2an 494 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → Fun (𝐹(1st ↾ (V × {𝐶}))))
8 dmco 5612 . . . . 5 dom (𝐹(1st ↾ (V × {𝐶}))) = ((1st ↾ (V × {𝐶})) “ dom 𝐹)
9 fndm 5958 . . . . . . . 8 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
109adantr 481 . . . . . . 7 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → dom 𝐹 = (𝐴 × 𝐵))
1110imaeq2d 5435 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((1st ↾ (V × {𝐶})) “ dom 𝐹) = ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)))
12 imacnvcnv 5568 . . . . . . . . 9 ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵))
13 df-ima 5097 . . . . . . . . 9 ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ran ((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵))
14 resres 5378 . . . . . . . . . 10 ((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵)) = (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵)))
1514rneqi 5322 . . . . . . . . 9 ran ((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵)) = ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵)))
1612, 13, 153eqtri 2647 . . . . . . . 8 ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵)))
17 inxp 5224 . . . . . . . . . . . . 13 ((V × {𝐶}) ∩ (𝐴 × 𝐵)) = ((V ∩ 𝐴) × ({𝐶} ∩ 𝐵))
18 incom 3789 . . . . . . . . . . . . . . 15 (V ∩ 𝐴) = (𝐴 ∩ V)
19 inv1 3948 . . . . . . . . . . . . . . 15 (𝐴 ∩ V) = 𝐴
2018, 19eqtri 2643 . . . . . . . . . . . . . 14 (V ∩ 𝐴) = 𝐴
2120xpeq1i 5105 . . . . . . . . . . . . 13 ((V ∩ 𝐴) × ({𝐶} ∩ 𝐵)) = (𝐴 × ({𝐶} ∩ 𝐵))
2217, 21eqtri 2643 . . . . . . . . . . . 12 ((V × {𝐶}) ∩ (𝐴 × 𝐵)) = (𝐴 × ({𝐶} ∩ 𝐵))
23 snssi 4315 . . . . . . . . . . . . . 14 (𝐶𝐵 → {𝐶} ⊆ 𝐵)
24 df-ss 3574 . . . . . . . . . . . . . 14 ({𝐶} ⊆ 𝐵 ↔ ({𝐶} ∩ 𝐵) = {𝐶})
2523, 24sylib 208 . . . . . . . . . . . . 13 (𝐶𝐵 → ({𝐶} ∩ 𝐵) = {𝐶})
2625xpeq2d 5109 . . . . . . . . . . . 12 (𝐶𝐵 → (𝐴 × ({𝐶} ∩ 𝐵)) = (𝐴 × {𝐶}))
2722, 26syl5eq 2667 . . . . . . . . . . 11 (𝐶𝐵 → ((V × {𝐶}) ∩ (𝐴 × 𝐵)) = (𝐴 × {𝐶}))
2827reseq2d 5366 . . . . . . . . . 10 (𝐶𝐵 → (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵))) = (1st ↾ (𝐴 × {𝐶})))
2928rneqd 5323 . . . . . . . . 9 (𝐶𝐵 → ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵))) = ran (1st ↾ (𝐴 × {𝐶})))
30 1stconst 7225 . . . . . . . . . 10 (𝐶𝐵 → (1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–1-1-onto𝐴)
31 f1ofo 6111 . . . . . . . . . 10 ((1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–1-1-onto𝐴 → (1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–onto𝐴)
32 forn 6085 . . . . . . . . . 10 ((1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–onto𝐴 → ran (1st ↾ (𝐴 × {𝐶})) = 𝐴)
3330, 31, 323syl 18 . . . . . . . . 9 (𝐶𝐵 → ran (1st ↾ (𝐴 × {𝐶})) = 𝐴)
3429, 33eqtrd 2655 . . . . . . . 8 (𝐶𝐵 → ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵))) = 𝐴)
3516, 34syl5eq 2667 . . . . . . 7 (𝐶𝐵 → ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = 𝐴)
3635adantl 482 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = 𝐴)
3711, 36eqtrd 2655 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((1st ↾ (V × {𝐶})) “ dom 𝐹) = 𝐴)
388, 37syl5eq 2667 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → dom (𝐹(1st ↾ (V × {𝐶}))) = 𝐴)
39 curry2.1 . . . . . 6 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
4039fneq1i 5953 . . . . 5 (𝐺 Fn 𝐴 ↔ (𝐹(1st ↾ (V × {𝐶}))) Fn 𝐴)
41 df-fn 5860 . . . . 5 ((𝐹(1st ↾ (V × {𝐶}))) Fn 𝐴 ↔ (Fun (𝐹(1st ↾ (V × {𝐶}))) ∧ dom (𝐹(1st ↾ (V × {𝐶}))) = 𝐴))
4240, 41bitri 264 . . . 4 (𝐺 Fn 𝐴 ↔ (Fun (𝐹(1st ↾ (V × {𝐶}))) ∧ dom (𝐹(1st ↾ (V × {𝐶}))) = 𝐴))
437, 38, 42sylanbrc 697 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 Fn 𝐴)
44 dffn5 6208 . . 3 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4543, 44sylib 208 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4639fveq1i 6159 . . . . 5 (𝐺𝑥) = ((𝐹(1st ↾ (V × {𝐶})))‘𝑥)
47 dff1o4 6112 . . . . . . . . 9 ((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V ↔ ((1st ↾ (V × {𝐶})) Fn (V × {𝐶}) ∧ (1st ↾ (V × {𝐶})) Fn V))
482, 47sylib 208 . . . . . . . 8 (𝐶𝐵 → ((1st ↾ (V × {𝐶})) Fn (V × {𝐶}) ∧ (1st ↾ (V × {𝐶})) Fn V))
4948simprd 479 . . . . . . 7 (𝐶𝐵(1st ↾ (V × {𝐶})) Fn V)
50 vex 3193 . . . . . . 7 𝑥 ∈ V
51 fvco2 6240 . . . . . . 7 (((1st ↾ (V × {𝐶})) Fn V ∧ 𝑥 ∈ V) → ((𝐹(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
5249, 50, 51sylancl 693 . . . . . 6 (𝐶𝐵 → ((𝐹(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
5352ad2antlr 762 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → ((𝐹(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
5446, 53syl5eq 2667 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
552adantr 481 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → (1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V)
5650a1i 11 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴) → 𝑥 ∈ V)
57 snidg 4184 . . . . . . . . . . 11 (𝐶𝐵𝐶 ∈ {𝐶})
5857adantr 481 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴) → 𝐶 ∈ {𝐶})
59 opelxp 5116 . . . . . . . . . 10 (⟨𝑥, 𝐶⟩ ∈ (V × {𝐶}) ↔ (𝑥 ∈ V ∧ 𝐶 ∈ {𝐶}))
6056, 58, 59sylanbrc 697 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶}))
6155, 60jca 554 . . . . . . . 8 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V ∧ ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶})))
6250a1i 11 . . . . . . . . . . . 12 (𝐶𝐵𝑥 ∈ V)
6362, 57, 59sylanbrc 697 . . . . . . . . . . 11 (𝐶𝐵 → ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶}))
64 fvres 6174 . . . . . . . . . . 11 (⟨𝑥, 𝐶⟩ ∈ (V × {𝐶}) → ((1st ↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = (1st ‘⟨𝑥, 𝐶⟩))
6563, 64syl 17 . . . . . . . . . 10 (𝐶𝐵 → ((1st ↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = (1st ‘⟨𝑥, 𝐶⟩))
6665adantr 481 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = (1st ‘⟨𝑥, 𝐶⟩))
67 op1stg 7140 . . . . . . . . . 10 ((𝑥𝐴𝐶𝐵) → (1st ‘⟨𝑥, 𝐶⟩) = 𝑥)
6867ancoms 469 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → (1st ‘⟨𝑥, 𝐶⟩) = 𝑥)
6966, 68eqtrd 2655 . . . . . . . 8 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = 𝑥)
70 f1ocnvfv 6499 . . . . . . . 8 (((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V ∧ ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶})) → (((1st ↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = 𝑥 → ((1st ↾ (V × {𝐶}))‘𝑥) = ⟨𝑥, 𝐶⟩))
7161, 69, 70sylc 65 . . . . . . 7 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶}))‘𝑥) = ⟨𝑥, 𝐶⟩)
7271fveq2d 6162 . . . . . 6 ((𝐶𝐵𝑥𝐴) → (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)) = (𝐹‘⟨𝑥, 𝐶⟩))
7372adantll 749 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)) = (𝐹‘⟨𝑥, 𝐶⟩))
74 df-ov 6618 . . . . 5 (𝑥𝐹𝐶) = (𝐹‘⟨𝑥, 𝐶⟩)
7573, 74syl6eqr 2673 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)) = (𝑥𝐹𝐶))
7654, 75eqtrd 2655 . . 3 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝑥𝐹𝐶))
7776mpteq2dva 4714 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝑥𝐴 ↦ (𝐺𝑥)) = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
7845, 77eqtrd 2655 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3190   ∩ cin 3559   ⊆ wss 3560  {csn 4155  ⟨cop 4161   ↦ cmpt 4683   × cxp 5082  ◡ccnv 5083  dom cdm 5084  ran crn 5085   ↾ cres 5086   “ cima 5087   ∘ ccom 5088  Fun wfun 5851   Fn wfn 5852  –onto→wfo 5855  –1-1-onto→wf1o 5856  ‘cfv 5857  (class class class)co 6615  1st c1st 7126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-1st 7128  df-2nd 7129 This theorem is referenced by:  curry2f  7233  curry2val  7234
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