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Theorem curry1 8127
Description: Composition with (2nd ↾ ({𝐶} × V)) turns any binary operation 𝐹 with a constant first operand into a function 𝐺 of the second operand only. This transformation is called "currying". (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)
Hypothesis
Ref Expression
curry1.1 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
Assertion
Ref Expression
curry1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝐺

Proof of Theorem curry1
StepHypRef Expression
1 fnfun 6668 . . . . 5 (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹)
2 2ndconst 8124 . . . . . 6 (𝐶𝐴 → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
3 dff1o3 6854 . . . . . . 7 ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–onto→V ∧ Fun (2nd ↾ ({𝐶} × V))))
43simprbi 496 . . . . . 6 ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V → Fun (2nd ↾ ({𝐶} × V)))
52, 4syl 17 . . . . 5 (𝐶𝐴 → Fun (2nd ↾ ({𝐶} × V)))
6 funco 6607 . . . . 5 ((Fun 𝐹 ∧ Fun (2nd ↾ ({𝐶} × V))) → Fun (𝐹(2nd ↾ ({𝐶} × V))))
71, 5, 6syl2an 596 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → Fun (𝐹(2nd ↾ ({𝐶} × V))))
8 dmco 6275 . . . . 5 dom (𝐹(2nd ↾ ({𝐶} × V))) = ((2nd ↾ ({𝐶} × V)) “ dom 𝐹)
9 fndm 6671 . . . . . . . 8 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
109adantr 480 . . . . . . 7 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → dom 𝐹 = (𝐴 × 𝐵))
1110imaeq2d 6079 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((2nd ↾ ({𝐶} × V)) “ dom 𝐹) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)))
12 imacnvcnv 6227 . . . . . . . . 9 ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵))
13 df-ima 5701 . . . . . . . . 9 ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵))
14 resres 6012 . . . . . . . . . 10 ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
1514rneqi 5950 . . . . . . . . 9 ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
1612, 13, 153eqtri 2766 . . . . . . . 8 ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
17 inxp 5844 . . . . . . . . . . . . 13 (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × (V ∩ 𝐵))
18 incom 4216 . . . . . . . . . . . . . . 15 (V ∩ 𝐵) = (𝐵 ∩ V)
19 inv1 4403 . . . . . . . . . . . . . . 15 (𝐵 ∩ V) = 𝐵
2018, 19eqtri 2762 . . . . . . . . . . . . . 14 (V ∩ 𝐵) = 𝐵
2120xpeq2i 5715 . . . . . . . . . . . . 13 (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
2217, 21eqtri 2762 . . . . . . . . . . . 12 (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
23 snssi 4812 . . . . . . . . . . . . . 14 (𝐶𝐴 → {𝐶} ⊆ 𝐴)
24 dfss2 3980 . . . . . . . . . . . . . 14 ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∩ 𝐴) = {𝐶})
2523, 24sylib 218 . . . . . . . . . . . . 13 (𝐶𝐴 → ({𝐶} ∩ 𝐴) = {𝐶})
2625xpeq1d 5717 . . . . . . . . . . . 12 (𝐶𝐴 → (({𝐶} ∩ 𝐴) × 𝐵) = ({𝐶} × 𝐵))
2722, 26eqtrid 2786 . . . . . . . . . . 11 (𝐶𝐴 → (({𝐶} × V) ∩ (𝐴 × 𝐵)) = ({𝐶} × 𝐵))
2827reseq2d 5999 . . . . . . . . . 10 (𝐶𝐴 → (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ ({𝐶} × 𝐵)))
2928rneqd 5951 . . . . . . . . 9 (𝐶𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = ran (2nd ↾ ({𝐶} × 𝐵)))
30 2ndconst 8124 . . . . . . . . . 10 (𝐶𝐴 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto𝐵)
31 f1ofo 6855 . . . . . . . . . 10 ((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto𝐵 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto𝐵)
32 forn 6823 . . . . . . . . . 10 ((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto𝐵 → ran (2nd ↾ ({𝐶} × 𝐵)) = 𝐵)
3330, 31, 323syl 18 . . . . . . . . 9 (𝐶𝐴 → ran (2nd ↾ ({𝐶} × 𝐵)) = 𝐵)
3429, 33eqtrd 2774 . . . . . . . 8 (𝐶𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = 𝐵)
3516, 34eqtrid 2786 . . . . . . 7 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
3635adantl 481 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
3711, 36eqtrd 2774 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((2nd ↾ ({𝐶} × V)) “ dom 𝐹) = 𝐵)
388, 37eqtrid 2786 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → dom (𝐹(2nd ↾ ({𝐶} × V))) = 𝐵)
39 curry1.1 . . . . . 6 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
4039fneq1i 6665 . . . . 5 (𝐺 Fn 𝐵 ↔ (𝐹(2nd ↾ ({𝐶} × V))) Fn 𝐵)
41 df-fn 6565 . . . . 5 ((𝐹(2nd ↾ ({𝐶} × V))) Fn 𝐵 ↔ (Fun (𝐹(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹(2nd ↾ ({𝐶} × V))) = 𝐵))
4240, 41bitri 275 . . . 4 (𝐺 Fn 𝐵 ↔ (Fun (𝐹(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹(2nd ↾ ({𝐶} × V))) = 𝐵))
437, 38, 42sylanbrc 583 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 Fn 𝐵)
44 dffn5 6966 . . 3 (𝐺 Fn 𝐵𝐺 = (𝑥𝐵 ↦ (𝐺𝑥)))
4543, 44sylib 218 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐺𝑥)))
4639fveq1i 6907 . . . . 5 (𝐺𝑥) = ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥)
47 dff1o4 6856 . . . . . . . 8 ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ (2nd ↾ ({𝐶} × V)) Fn V))
482, 47sylib 218 . . . . . . 7 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ (2nd ↾ ({𝐶} × V)) Fn V))
49 fvco2 7005 . . . . . . . 8 (((2nd ↾ ({𝐶} × V)) Fn V ∧ 𝑥 ∈ V) → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5049elvd 3483 . . . . . . 7 ((2nd ↾ ({𝐶} × V)) Fn V → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5148, 50simpl2im 503 . . . . . 6 (𝐶𝐴 → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5251ad2antlr 727 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5346, 52eqtrid 2786 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
542adantr 480 . . . . . . . . 9 ((𝐶𝐴𝑥𝐵) → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
55 snidg 4664 . . . . . . . . . . 11 (𝐶𝐴𝐶 ∈ {𝐶})
56 vex 3481 . . . . . . . . . . 11 𝑥 ∈ V
57 opelxp 5724 . . . . . . . . . . 11 (⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V) ↔ (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V))
5855, 56, 57sylanblrc 590 . . . . . . . . . 10 (𝐶𝐴 → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
5958adantr 480 . . . . . . . . 9 ((𝐶𝐴𝑥𝐵) → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
6054, 59jca 511 . . . . . . . 8 ((𝐶𝐴𝑥𝐵) → ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ∧ ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V)))
6158fvresd 6926 . . . . . . . . . 10 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = (2nd ‘⟨𝐶, 𝑥⟩))
62 op2ndg 8025 . . . . . . . . . . 11 ((𝐶𝐴𝑥 ∈ V) → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
6362elvd 3483 . . . . . . . . . 10 (𝐶𝐴 → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
6461, 63eqtrd 2774 . . . . . . . . 9 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
6564adantr 480 . . . . . . . 8 ((𝐶𝐴𝑥𝐵) → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
66 f1ocnvfv 7297 . . . . . . . 8 (((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ∧ ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V)) → (((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥 → ((2nd ↾ ({𝐶} × V))‘𝑥) = ⟨𝐶, 𝑥⟩))
6760, 65, 66sylc 65 . . . . . . 7 ((𝐶𝐴𝑥𝐵) → ((2nd ↾ ({𝐶} × V))‘𝑥) = ⟨𝐶, 𝑥⟩)
6867fveq2d 6910 . . . . . 6 ((𝐶𝐴𝑥𝐵) → (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
6968adantll 714 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
70 df-ov 7433 . . . . 5 (𝐶𝐹𝑥) = (𝐹‘⟨𝐶, 𝑥⟩)
7169, 70eqtr4di 2792 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐶𝐹𝑥))
7253, 71eqtrd 2774 . . 3 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐶𝐹𝑥))
7372mpteq2dva 5247 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (𝑥𝐵 ↦ (𝐺𝑥)) = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
7445, 73eqtrd 2774 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  cin 3961  wss 3962  {csn 4630  cop 4636  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  ccom 5692  Fun wfun 6556   Fn wfn 6557  ontowfo 6560  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  2nd c2nd 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-1st 8012  df-2nd 8013
This theorem is referenced by:  curry1val  8128  curry1f  8129
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