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Theorem curry1 7420
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 6128 . . . . 5 (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹)
2 2ndconst 7417 . . . . . 6 (𝐶𝐴 → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
3 dff1o3 6284 . . . . . . 7 ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–onto→V ∧ Fun (2nd ↾ ({𝐶} × V))))
43simprbi 484 . . . . . 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 6071 . . . . 5 ((Fun 𝐹 ∧ Fun (2nd ↾ ({𝐶} × V))) → Fun (𝐹(2nd ↾ ({𝐶} × V))))
71, 5, 6syl2an 583 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → Fun (𝐹(2nd ↾ ({𝐶} × V))))
8 dmco 5787 . . . . 5 dom (𝐹(2nd ↾ ({𝐶} × V))) = ((2nd ↾ ({𝐶} × V)) “ dom 𝐹)
9 fndm 6130 . . . . . . . 8 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
109adantr 466 . . . . . . 7 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → dom 𝐹 = (𝐴 × 𝐵))
1110imaeq2d 5607 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((2nd ↾ ({𝐶} × V)) “ dom 𝐹) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)))
12 imacnvcnv 5740 . . . . . . . . 9 ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵))
13 df-ima 5262 . . . . . . . . 9 ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵))
14 resres 5550 . . . . . . . . . 10 ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
1514rneqi 5490 . . . . . . . . 9 ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
1612, 13, 153eqtri 2797 . . . . . . . 8 ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
17 inxp 5393 . . . . . . . . . . . . 13 (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × (V ∩ 𝐵))
18 incom 3956 . . . . . . . . . . . . . . 15 (V ∩ 𝐵) = (𝐵 ∩ V)
19 inv1 4114 . . . . . . . . . . . . . . 15 (𝐵 ∩ V) = 𝐵
2018, 19eqtri 2793 . . . . . . . . . . . . . 14 (V ∩ 𝐵) = 𝐵
2120xpeq2i 5276 . . . . . . . . . . . . 13 (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
2217, 21eqtri 2793 . . . . . . . . . . . 12 (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
23 snssi 4474 . . . . . . . . . . . . . 14 (𝐶𝐴 → {𝐶} ⊆ 𝐴)
24 df-ss 3737 . . . . . . . . . . . . . 14 ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∩ 𝐴) = {𝐶})
2523, 24sylib 208 . . . . . . . . . . . . 13 (𝐶𝐴 → ({𝐶} ∩ 𝐴) = {𝐶})
2625xpeq1d 5278 . . . . . . . . . . . 12 (𝐶𝐴 → (({𝐶} ∩ 𝐴) × 𝐵) = ({𝐶} × 𝐵))
2722, 26syl5eq 2817 . . . . . . . . . . 11 (𝐶𝐴 → (({𝐶} × V) ∩ (𝐴 × 𝐵)) = ({𝐶} × 𝐵))
2827reseq2d 5534 . . . . . . . . . 10 (𝐶𝐴 → (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ ({𝐶} × 𝐵)))
2928rneqd 5491 . . . . . . . . 9 (𝐶𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = ran (2nd ↾ ({𝐶} × 𝐵)))
30 2ndconst 7417 . . . . . . . . . 10 (𝐶𝐴 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto𝐵)
31 f1ofo 6285 . . . . . . . . . 10 ((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto𝐵 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto𝐵)
32 forn 6259 . . . . . . . . . 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 2805 . . . . . . . 8 (𝐶𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = 𝐵)
3516, 34syl5eq 2817 . . . . . . 7 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
3635adantl 467 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
3711, 36eqtrd 2805 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((2nd ↾ ({𝐶} × V)) “ dom 𝐹) = 𝐵)
388, 37syl5eq 2817 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → dom (𝐹(2nd ↾ ({𝐶} × V))) = 𝐵)
39 curry1.1 . . . . . 6 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
4039fneq1i 6125 . . . . 5 (𝐺 Fn 𝐵 ↔ (𝐹(2nd ↾ ({𝐶} × V))) Fn 𝐵)
41 df-fn 6034 . . . . 5 ((𝐹(2nd ↾ ({𝐶} × V))) Fn 𝐵 ↔ (Fun (𝐹(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹(2nd ↾ ({𝐶} × V))) = 𝐵))
4240, 41bitri 264 . . . 4 (𝐺 Fn 𝐵 ↔ (Fun (𝐹(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹(2nd ↾ ({𝐶} × V))) = 𝐵))
437, 38, 42sylanbrc 572 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 Fn 𝐵)
44 dffn5 6383 . . 3 (𝐺 Fn 𝐵𝐺 = (𝑥𝐵 ↦ (𝐺𝑥)))
4543, 44sylib 208 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐺𝑥)))
4639fveq1i 6333 . . . . 5 (𝐺𝑥) = ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥)
47 dff1o4 6286 . . . . . . . . 9 ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ (2nd ↾ ({𝐶} × V)) Fn V))
482, 47sylib 208 . . . . . . . 8 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ (2nd ↾ ({𝐶} × V)) Fn V))
4948simprd 483 . . . . . . 7 (𝐶𝐴(2nd ↾ ({𝐶} × V)) Fn V)
50 vex 3354 . . . . . . . 8 𝑥 ∈ V
51 fvco2 6415 . . . . . . . 8 (((2nd ↾ ({𝐶} × V)) Fn V ∧ 𝑥 ∈ V) → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5250, 51mpan2 671 . . . . . . 7 ((2nd ↾ ({𝐶} × V)) Fn V → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5349, 52syl 17 . . . . . 6 (𝐶𝐴 → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5453ad2antlr 706 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5546, 54syl5eq 2817 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
562adantr 466 . . . . . . . . 9 ((𝐶𝐴𝑥𝐵) → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
57 snidg 4345 . . . . . . . . . . . 12 (𝐶𝐴𝐶 ∈ {𝐶})
5857, 50jctir 510 . . . . . . . . . . 11 (𝐶𝐴 → (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V))
59 opelxp 5286 . . . . . . . . . . 11 (⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V) ↔ (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V))
6058, 59sylibr 224 . . . . . . . . . 10 (𝐶𝐴 → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
6160adantr 466 . . . . . . . . 9 ((𝐶𝐴𝑥𝐵) → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
6256, 61jca 501 . . . . . . . 8 ((𝐶𝐴𝑥𝐵) → ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ∧ ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V)))
63 fvres 6348 . . . . . . . . . . 11 (⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V) → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = (2nd ‘⟨𝐶, 𝑥⟩))
6460, 63syl 17 . . . . . . . . . 10 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = (2nd ‘⟨𝐶, 𝑥⟩))
65 op2ndg 7328 . . . . . . . . . . 11 ((𝐶𝐴𝑥 ∈ V) → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
6650, 65mpan2 671 . . . . . . . . . 10 (𝐶𝐴 → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
6764, 66eqtrd 2805 . . . . . . . . 9 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
6867adantr 466 . . . . . . . 8 ((𝐶𝐴𝑥𝐵) → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
69 f1ocnvfv 6677 . . . . . . . 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⟩))
7062, 68, 69sylc 65 . . . . . . 7 ((𝐶𝐴𝑥𝐵) → ((2nd ↾ ({𝐶} × V))‘𝑥) = ⟨𝐶, 𝑥⟩)
7170fveq2d 6336 . . . . . 6 ((𝐶𝐴𝑥𝐵) → (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
7271adantll 693 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
73 df-ov 6796 . . . . 5 (𝐶𝐹𝑥) = (𝐹‘⟨𝐶, 𝑥⟩)
7472, 73syl6eqr 2823 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐶𝐹𝑥))
7555, 74eqtrd 2805 . . 3 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐶𝐹𝑥))
7675mpteq2dva 4878 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (𝑥𝐵 ↦ (𝐺𝑥)) = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
7745, 76eqtrd 2805 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cin 3722  wss 3723  {csn 4316  cop 4322  cmpt 4863   × cxp 5247  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  ccom 5253  Fun wfun 6025   Fn wfn 6026  ontowfo 6029  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  2nd c2nd 7314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-1st 7315  df-2nd 7316
This theorem is referenced by:  curry1val  7421  curry1f  7422
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