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Theorem curry2 7798
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 6452 . . . . 5 (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹)
2 1stconst 7791 . . . . . 6 (𝐶𝐵 → (1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V)
3 dff1o3 6620 . . . . . . 7 ((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V ↔ ((1st ↾ (V × {𝐶})):(V × {𝐶})–onto→V ∧ Fun (1st ↾ (V × {𝐶}))))
43simprbi 497 . . . . . 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 6394 . . . . 5 ((Fun 𝐹 ∧ Fun (1st ↾ (V × {𝐶}))) → Fun (𝐹(1st ↾ (V × {𝐶}))))
71, 5, 6syl2an 595 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → Fun (𝐹(1st ↾ (V × {𝐶}))))
8 dmco 6106 . . . . 5 dom (𝐹(1st ↾ (V × {𝐶}))) = ((1st ↾ (V × {𝐶})) “ dom 𝐹)
9 fndm 6454 . . . . . . . 8 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
109adantr 481 . . . . . . 7 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → dom 𝐹 = (𝐴 × 𝐵))
1110imaeq2d 5928 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((1st ↾ (V × {𝐶})) “ dom 𝐹) = ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)))
12 imacnvcnv 6062 . . . . . . . . 9 ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵))
13 df-ima 5567 . . . . . . . . 9 ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ran ((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵))
14 resres 5865 . . . . . . . . . 10 ((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵)) = (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵)))
1514rneqi 5806 . . . . . . . . 9 ran ((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵)) = ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵)))
1612, 13, 153eqtri 2853 . . . . . . . 8 ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵)))
17 inxp 5702 . . . . . . . . . . . . 13 ((V × {𝐶}) ∩ (𝐴 × 𝐵)) = ((V ∩ 𝐴) × ({𝐶} ∩ 𝐵))
18 incom 4182 . . . . . . . . . . . . . . 15 (V ∩ 𝐴) = (𝐴 ∩ V)
19 inv1 4352 . . . . . . . . . . . . . . 15 (𝐴 ∩ V) = 𝐴
2018, 19eqtri 2849 . . . . . . . . . . . . . 14 (V ∩ 𝐴) = 𝐴
2120xpeq1i 5580 . . . . . . . . . . . . 13 ((V ∩ 𝐴) × ({𝐶} ∩ 𝐵)) = (𝐴 × ({𝐶} ∩ 𝐵))
2217, 21eqtri 2849 . . . . . . . . . . . 12 ((V × {𝐶}) ∩ (𝐴 × 𝐵)) = (𝐴 × ({𝐶} ∩ 𝐵))
23 snssi 4740 . . . . . . . . . . . . . 14 (𝐶𝐵 → {𝐶} ⊆ 𝐵)
24 df-ss 3956 . . . . . . . . . . . . . 14 ({𝐶} ⊆ 𝐵 ↔ ({𝐶} ∩ 𝐵) = {𝐶})
2523, 24sylib 219 . . . . . . . . . . . . 13 (𝐶𝐵 → ({𝐶} ∩ 𝐵) = {𝐶})
2625xpeq2d 5584 . . . . . . . . . . . 12 (𝐶𝐵 → (𝐴 × ({𝐶} ∩ 𝐵)) = (𝐴 × {𝐶}))
2722, 26syl5eq 2873 . . . . . . . . . . 11 (𝐶𝐵 → ((V × {𝐶}) ∩ (𝐴 × 𝐵)) = (𝐴 × {𝐶}))
2827reseq2d 5852 . . . . . . . . . 10 (𝐶𝐵 → (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵))) = (1st ↾ (𝐴 × {𝐶})))
2928rneqd 5807 . . . . . . . . 9 (𝐶𝐵 → ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵))) = ran (1st ↾ (𝐴 × {𝐶})))
30 1stconst 7791 . . . . . . . . . 10 (𝐶𝐵 → (1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–1-1-onto𝐴)
31 f1ofo 6621 . . . . . . . . . 10 ((1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–1-1-onto𝐴 → (1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–onto𝐴)
32 forn 6592 . . . . . . . . . 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 2861 . . . . . . . 8 (𝐶𝐵 → ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵))) = 𝐴)
3516, 34syl5eq 2873 . . . . . . 7 (𝐶𝐵 → ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = 𝐴)
3635adantl 482 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = 𝐴)
3711, 36eqtrd 2861 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((1st ↾ (V × {𝐶})) “ dom 𝐹) = 𝐴)
388, 37syl5eq 2873 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → dom (𝐹(1st ↾ (V × {𝐶}))) = 𝐴)
39 curry2.1 . . . . . 6 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
4039fneq1i 6449 . . . . 5 (𝐺 Fn 𝐴 ↔ (𝐹(1st ↾ (V × {𝐶}))) Fn 𝐴)
41 df-fn 6357 . . . . 5 ((𝐹(1st ↾ (V × {𝐶}))) Fn 𝐴 ↔ (Fun (𝐹(1st ↾ (V × {𝐶}))) ∧ dom (𝐹(1st ↾ (V × {𝐶}))) = 𝐴))
4240, 41bitri 276 . . . 4 (𝐺 Fn 𝐴 ↔ (Fun (𝐹(1st ↾ (V × {𝐶}))) ∧ dom (𝐹(1st ↾ (V × {𝐶}))) = 𝐴))
437, 38, 42sylanbrc 583 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 Fn 𝐴)
44 dffn5 6723 . . 3 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4543, 44sylib 219 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4639fveq1i 6670 . . . . 5 (𝐺𝑥) = ((𝐹(1st ↾ (V × {𝐶})))‘𝑥)
47 dff1o4 6622 . . . . . . . . 9 ((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V ↔ ((1st ↾ (V × {𝐶})) Fn (V × {𝐶}) ∧ (1st ↾ (V × {𝐶})) Fn V))
482, 47sylib 219 . . . . . . . 8 (𝐶𝐵 → ((1st ↾ (V × {𝐶})) Fn (V × {𝐶}) ∧ (1st ↾ (V × {𝐶})) Fn V))
4948simprd 496 . . . . . . 7 (𝐶𝐵(1st ↾ (V × {𝐶})) Fn V)
50 vex 3503 . . . . . . 7 𝑥 ∈ V
51 fvco2 6757 . . . . . . 7 (((1st ↾ (V × {𝐶})) Fn V ∧ 𝑥 ∈ V) → ((𝐹(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
5249, 50, 51sylancl 586 . . . . . 6 (𝐶𝐵 → ((𝐹(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
5352ad2antlr 723 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → ((𝐹(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
5446, 53syl5eq 2873 . . . 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 4596 . . . . . . . . . . 11 (𝐶𝐵𝐶 ∈ {𝐶})
5857adantr 481 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴) → 𝐶 ∈ {𝐶})
5956, 58opelxpd 5592 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶}))
6055, 59jca 512 . . . . . . . 8 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V ∧ ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶})))
6150a1i 11 . . . . . . . . . . . 12 (𝐶𝐵𝑥 ∈ V)
6261, 57opelxpd 5592 . . . . . . . . . . 11 (𝐶𝐵 → ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶}))
6362fvresd 6689 . . . . . . . . . 10 (𝐶𝐵 → ((1st ↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = (1st ‘⟨𝑥, 𝐶⟩))
6463adantr 481 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = (1st ‘⟨𝑥, 𝐶⟩))
65 op1stg 7697 . . . . . . . . . 10 ((𝑥𝐴𝐶𝐵) → (1st ‘⟨𝑥, 𝐶⟩) = 𝑥)
6665ancoms 459 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → (1st ‘⟨𝑥, 𝐶⟩) = 𝑥)
6764, 66eqtrd 2861 . . . . . . . 8 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = 𝑥)
68 f1ocnvfv 7031 . . . . . . . 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⟩))
6960, 67, 68sylc 65 . . . . . . 7 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶}))‘𝑥) = ⟨𝑥, 𝐶⟩)
7069fveq2d 6673 . . . . . 6 ((𝐶𝐵𝑥𝐴) → (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)) = (𝐹‘⟨𝑥, 𝐶⟩))
7170adantll 710 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)) = (𝐹‘⟨𝑥, 𝐶⟩))
72 df-ov 7153 . . . . 5 (𝑥𝐹𝐶) = (𝐹‘⟨𝑥, 𝐶⟩)
7371, 72syl6eqr 2879 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)) = (𝑥𝐹𝐶))
7454, 73eqtrd 2861 . . 3 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝑥𝐹𝐶))
7574mpteq2dva 5158 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝑥𝐴 ↦ (𝐺𝑥)) = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
7645, 75eqtrd 2861 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  cin 3939  wss 3940  {csn 4564  cop 4570  cmpt 5143   × cxp 5552  ccnv 5553  dom cdm 5554  ran crn 5555  cres 5556  cima 5557  ccom 5558  Fun wfun 6348   Fn wfn 6349  ontowfo 6352  1-1-ontowf1o 6353  cfv 6354  (class class class)co 7150  1st c1st 7683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-1st 7685  df-2nd 7686
This theorem is referenced by:  curry2f  7799  curry2val  7800
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