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Theorem curry1 7503
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 6199 . . . . 5 (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹)
2 2ndconst 7500 . . . . . 6 (𝐶𝐴 → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
3 dff1o3 6359 . . . . . . 7 ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–onto→V ∧ Fun (2nd ↾ ({𝐶} × V))))
43simprbi 486 . . . . . 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 6141 . . . . 5 ((Fun 𝐹 ∧ Fun (2nd ↾ ({𝐶} × V))) → Fun (𝐹(2nd ↾ ({𝐶} × V))))
71, 5, 6syl2an 585 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → Fun (𝐹(2nd ↾ ({𝐶} × V))))
8 dmco 5857 . . . . 5 dom (𝐹(2nd ↾ ({𝐶} × V))) = ((2nd ↾ ({𝐶} × V)) “ dom 𝐹)
9 fndm 6201 . . . . . . . 8 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
109adantr 468 . . . . . . 7 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → dom 𝐹 = (𝐴 × 𝐵))
1110imaeq2d 5676 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((2nd ↾ ({𝐶} × V)) “ dom 𝐹) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)))
12 imacnvcnv 5810 . . . . . . . . 9 ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵))
13 df-ima 5324 . . . . . . . . 9 ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵))
14 resres 5613 . . . . . . . . . 10 ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
1514rneqi 5553 . . . . . . . . 9 ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
1612, 13, 153eqtri 2832 . . . . . . . 8 ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
17 inxp 5456 . . . . . . . . . . . . 13 (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × (V ∩ 𝐵))
18 incom 4004 . . . . . . . . . . . . . . 15 (V ∩ 𝐵) = (𝐵 ∩ V)
19 inv1 4168 . . . . . . . . . . . . . . 15 (𝐵 ∩ V) = 𝐵
2018, 19eqtri 2828 . . . . . . . . . . . . . 14 (V ∩ 𝐵) = 𝐵
2120xpeq2i 5337 . . . . . . . . . . . . 13 (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
2217, 21eqtri 2828 . . . . . . . . . . . 12 (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
23 snssi 4529 . . . . . . . . . . . . . 14 (𝐶𝐴 → {𝐶} ⊆ 𝐴)
24 df-ss 3783 . . . . . . . . . . . . . 14 ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∩ 𝐴) = {𝐶})
2523, 24sylib 209 . . . . . . . . . . . . 13 (𝐶𝐴 → ({𝐶} ∩ 𝐴) = {𝐶})
2625xpeq1d 5339 . . . . . . . . . . . 12 (𝐶𝐴 → (({𝐶} ∩ 𝐴) × 𝐵) = ({𝐶} × 𝐵))
2722, 26syl5eq 2852 . . . . . . . . . . 11 (𝐶𝐴 → (({𝐶} × V) ∩ (𝐴 × 𝐵)) = ({𝐶} × 𝐵))
2827reseq2d 5597 . . . . . . . . . 10 (𝐶𝐴 → (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ ({𝐶} × 𝐵)))
2928rneqd 5554 . . . . . . . . 9 (𝐶𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = ran (2nd ↾ ({𝐶} × 𝐵)))
30 2ndconst 7500 . . . . . . . . . 10 (𝐶𝐴 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto𝐵)
31 f1ofo 6360 . . . . . . . . . 10 ((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto𝐵 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto𝐵)
32 forn 6334 . . . . . . . . . 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 2840 . . . . . . . 8 (𝐶𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = 𝐵)
3516, 34syl5eq 2852 . . . . . . 7 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
3635adantl 469 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
3711, 36eqtrd 2840 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → ((2nd ↾ ({𝐶} × V)) “ dom 𝐹) = 𝐵)
388, 37syl5eq 2852 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → dom (𝐹(2nd ↾ ({𝐶} × V))) = 𝐵)
39 curry1.1 . . . . . 6 𝐺 = (𝐹(2nd ↾ ({𝐶} × V)))
4039fneq1i 6196 . . . . 5 (𝐺 Fn 𝐵 ↔ (𝐹(2nd ↾ ({𝐶} × V))) Fn 𝐵)
41 df-fn 6104 . . . . 5 ((𝐹(2nd ↾ ({𝐶} × V))) Fn 𝐵 ↔ (Fun (𝐹(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹(2nd ↾ ({𝐶} × V))) = 𝐵))
4240, 41bitri 266 . . . 4 (𝐺 Fn 𝐵 ↔ (Fun (𝐹(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹(2nd ↾ ({𝐶} × V))) = 𝐵))
437, 38, 42sylanbrc 574 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 Fn 𝐵)
44 dffn5 6462 . . 3 (𝐺 Fn 𝐵𝐺 = (𝑥𝐵 ↦ (𝐺𝑥)))
4543, 44sylib 209 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐺𝑥)))
4639fveq1i 6409 . . . . 5 (𝐺𝑥) = ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥)
47 dff1o4 6361 . . . . . . . . 9 ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ (2nd ↾ ({𝐶} × V)) Fn V))
482, 47sylib 209 . . . . . . . 8 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ (2nd ↾ ({𝐶} × V)) Fn V))
4948simprd 485 . . . . . . 7 (𝐶𝐴(2nd ↾ ({𝐶} × V)) Fn V)
50 fvco2 6494 . . . . . . . 8 (((2nd ↾ ({𝐶} × V)) Fn V ∧ 𝑥 ∈ V) → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5150elvd 3396 . . . . . . 7 ((2nd ↾ ({𝐶} × V)) Fn V → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5249, 51syl 17 . . . . . 6 (𝐶𝐴 → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5352ad2antlr 709 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → ((𝐹(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
5446, 53syl5eq 2852 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)))
552adantr 468 . . . . . . . . 9 ((𝐶𝐴𝑥𝐵) → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
56 snidg 4400 . . . . . . . . . . . 12 (𝐶𝐴𝐶 ∈ {𝐶})
57 vex 3394 . . . . . . . . . . . 12 𝑥 ∈ V
5856, 57jctir 512 . . . . . . . . . . 11 (𝐶𝐴 → (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V))
59 opelxp 5346 . . . . . . . . . . 11 (⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V) ↔ (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V))
6058, 59sylibr 225 . . . . . . . . . 10 (𝐶𝐴 → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
6160adantr 468 . . . . . . . . 9 ((𝐶𝐴𝑥𝐵) → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
6255, 61jca 503 . . . . . . . 8 ((𝐶𝐴𝑥𝐵) → ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ∧ ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V)))
63 fvres 6427 . . . . . . . . . . 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 7411 . . . . . . . . . . 11 ((𝐶𝐴𝑥 ∈ V) → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
6665elvd 3396 . . . . . . . . . 10 (𝐶𝐴 → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
6764, 66eqtrd 2840 . . . . . . . . 9 (𝐶𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
6867adantr 468 . . . . . . . 8 ((𝐶𝐴𝑥𝐵) → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
69 f1ocnvfv 6758 . . . . . . . 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 6412 . . . . . 6 ((𝐶𝐴𝑥𝐵) → (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
7271adantll 696 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
73 df-ov 6877 . . . . 5 (𝐶𝐹𝑥) = (𝐹‘⟨𝐶, 𝑥⟩)
7472, 73syl6eqr 2858 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐹‘((2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐶𝐹𝑥))
7554, 74eqtrd 2840 . . 3 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐶𝐹𝑥))
7675mpteq2dva 4938 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → (𝑥𝐵 ↦ (𝐺𝑥)) = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
7745, 76eqtrd 2840 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴) → 𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  Vcvv 3391  cin 3768  wss 3769  {csn 4370  cop 4376  cmpt 4923   × cxp 5309  ccnv 5310  dom cdm 5311  ran crn 5312  cres 5313  cima 5314  ccom 5315  Fun wfun 6095   Fn wfn 6096  ontowfo 6099  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6874  2nd c2nd 7397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-1st 7398  df-2nd 7399
This theorem is referenced by:  curry1val  7504  curry1f  7505
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