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Theorem curry2 7474
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 6166 . . . . 5 (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹)
2 1stconst 7467 . . . . . 6 (𝐶𝐵 → (1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V)
3 dff1o3 6326 . . . . . . 7 ((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V ↔ ((1st ↾ (V × {𝐶})):(V × {𝐶})–onto→V ∧ Fun (1st ↾ (V × {𝐶}))))
43simprbi 490 . . . . . 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 6108 . . . . 5 ((Fun 𝐹 ∧ Fun (1st ↾ (V × {𝐶}))) → Fun (𝐹(1st ↾ (V × {𝐶}))))
71, 5, 6syl2an 589 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → Fun (𝐹(1st ↾ (V × {𝐶}))))
8 dmco 5829 . . . . 5 dom (𝐹(1st ↾ (V × {𝐶}))) = ((1st ↾ (V × {𝐶})) “ dom 𝐹)
9 fndm 6168 . . . . . . . 8 (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
109adantr 472 . . . . . . 7 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → dom 𝐹 = (𝐴 × 𝐵))
1110imaeq2d 5648 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((1st ↾ (V × {𝐶})) “ dom 𝐹) = ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)))
12 imacnvcnv 5782 . . . . . . . . 9 ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵))
13 df-ima 5290 . . . . . . . . 9 ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ran ((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵))
14 resres 5585 . . . . . . . . . 10 ((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵)) = (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵)))
1514rneqi 5520 . . . . . . . . 9 ran ((1st ↾ (V × {𝐶})) ↾ (𝐴 × 𝐵)) = ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵)))
1612, 13, 153eqtri 2791 . . . . . . . 8 ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵)))
17 inxp 5423 . . . . . . . . . . . . 13 ((V × {𝐶}) ∩ (𝐴 × 𝐵)) = ((V ∩ 𝐴) × ({𝐶} ∩ 𝐵))
18 incom 3967 . . . . . . . . . . . . . . 15 (V ∩ 𝐴) = (𝐴 ∩ V)
19 inv1 4132 . . . . . . . . . . . . . . 15 (𝐴 ∩ V) = 𝐴
2018, 19eqtri 2787 . . . . . . . . . . . . . 14 (V ∩ 𝐴) = 𝐴
2120xpeq1i 5303 . . . . . . . . . . . . 13 ((V ∩ 𝐴) × ({𝐶} ∩ 𝐵)) = (𝐴 × ({𝐶} ∩ 𝐵))
2217, 21eqtri 2787 . . . . . . . . . . . 12 ((V × {𝐶}) ∩ (𝐴 × 𝐵)) = (𝐴 × ({𝐶} ∩ 𝐵))
23 snssi 4493 . . . . . . . . . . . . . 14 (𝐶𝐵 → {𝐶} ⊆ 𝐵)
24 df-ss 3746 . . . . . . . . . . . . . 14 ({𝐶} ⊆ 𝐵 ↔ ({𝐶} ∩ 𝐵) = {𝐶})
2523, 24sylib 209 . . . . . . . . . . . . 13 (𝐶𝐵 → ({𝐶} ∩ 𝐵) = {𝐶})
2625xpeq2d 5307 . . . . . . . . . . . 12 (𝐶𝐵 → (𝐴 × ({𝐶} ∩ 𝐵)) = (𝐴 × {𝐶}))
2722, 26syl5eq 2811 . . . . . . . . . . 11 (𝐶𝐵 → ((V × {𝐶}) ∩ (𝐴 × 𝐵)) = (𝐴 × {𝐶}))
2827reseq2d 5565 . . . . . . . . . 10 (𝐶𝐵 → (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵))) = (1st ↾ (𝐴 × {𝐶})))
2928rneqd 5521 . . . . . . . . 9 (𝐶𝐵 → ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵))) = ran (1st ↾ (𝐴 × {𝐶})))
30 1stconst 7467 . . . . . . . . . 10 (𝐶𝐵 → (1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–1-1-onto𝐴)
31 f1ofo 6327 . . . . . . . . . 10 ((1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–1-1-onto𝐴 → (1st ↾ (𝐴 × {𝐶})):(𝐴 × {𝐶})–onto𝐴)
32 forn 6301 . . . . . . . . . 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 2799 . . . . . . . 8 (𝐶𝐵 → ran (1st ↾ ((V × {𝐶}) ∩ (𝐴 × 𝐵))) = 𝐴)
3516, 34syl5eq 2811 . . . . . . 7 (𝐶𝐵 → ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = 𝐴)
3635adantl 473 . . . . . 6 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((1st ↾ (V × {𝐶})) “ (𝐴 × 𝐵)) = 𝐴)
3711, 36eqtrd 2799 . . . . 5 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → ((1st ↾ (V × {𝐶})) “ dom 𝐹) = 𝐴)
388, 37syl5eq 2811 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → dom (𝐹(1st ↾ (V × {𝐶}))) = 𝐴)
39 curry2.1 . . . . . 6 𝐺 = (𝐹(1st ↾ (V × {𝐶})))
4039fneq1i 6163 . . . . 5 (𝐺 Fn 𝐴 ↔ (𝐹(1st ↾ (V × {𝐶}))) Fn 𝐴)
41 df-fn 6071 . . . . 5 ((𝐹(1st ↾ (V × {𝐶}))) Fn 𝐴 ↔ (Fun (𝐹(1st ↾ (V × {𝐶}))) ∧ dom (𝐹(1st ↾ (V × {𝐶}))) = 𝐴))
4240, 41bitri 266 . . . 4 (𝐺 Fn 𝐴 ↔ (Fun (𝐹(1st ↾ (V × {𝐶}))) ∧ dom (𝐹(1st ↾ (V × {𝐶}))) = 𝐴))
437, 38, 42sylanbrc 578 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 Fn 𝐴)
44 dffn5 6430 . . 3 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4543, 44sylib 209 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4639fveq1i 6376 . . . . 5 (𝐺𝑥) = ((𝐹(1st ↾ (V × {𝐶})))‘𝑥)
47 dff1o4 6328 . . . . . . . . 9 ((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V ↔ ((1st ↾ (V × {𝐶})) Fn (V × {𝐶}) ∧ (1st ↾ (V × {𝐶})) Fn V))
482, 47sylib 209 . . . . . . . 8 (𝐶𝐵 → ((1st ↾ (V × {𝐶})) Fn (V × {𝐶}) ∧ (1st ↾ (V × {𝐶})) Fn V))
4948simprd 489 . . . . . . 7 (𝐶𝐵(1st ↾ (V × {𝐶})) Fn V)
50 vex 3353 . . . . . . 7 𝑥 ∈ V
51 fvco2 6462 . . . . . . 7 (((1st ↾ (V × {𝐶})) Fn V ∧ 𝑥 ∈ V) → ((𝐹(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
5249, 50, 51sylancl 580 . . . . . 6 (𝐶𝐵 → ((𝐹(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
5352ad2antlr 718 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → ((𝐹(1st ↾ (V × {𝐶})))‘𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
5446, 53syl5eq 2811 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)))
552adantr 472 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → (1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V)
5650a1i 11 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴) → 𝑥 ∈ V)
57 snidg 4364 . . . . . . . . . . 11 (𝐶𝐵𝐶 ∈ {𝐶})
5857adantr 472 . . . . . . . . . 10 ((𝐶𝐵𝑥𝐴) → 𝐶 ∈ {𝐶})
59 opelxp 5313 . . . . . . . . . 10 (⟨𝑥, 𝐶⟩ ∈ (V × {𝐶}) ↔ (𝑥 ∈ V ∧ 𝐶 ∈ {𝐶}))
6056, 58, 59sylanbrc 578 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶}))
6155, 60jca 507 . . . . . . . 8 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶})):(V × {𝐶})–1-1-onto→V ∧ ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶})))
6250a1i 11 . . . . . . . . . . . 12 (𝐶𝐵𝑥 ∈ V)
6362, 57, 59sylanbrc 578 . . . . . . . . . . 11 (𝐶𝐵 → ⟨𝑥, 𝐶⟩ ∈ (V × {𝐶}))
64 fvres 6394 . . . . . . . . . . 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 472 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = (1st ‘⟨𝑥, 𝐶⟩))
67 op1stg 7378 . . . . . . . . . 10 ((𝑥𝐴𝐶𝐵) → (1st ‘⟨𝑥, 𝐶⟩) = 𝑥)
6867ancoms 450 . . . . . . . . 9 ((𝐶𝐵𝑥𝐴) → (1st ‘⟨𝑥, 𝐶⟩) = 𝑥)
6966, 68eqtrd 2799 . . . . . . . 8 ((𝐶𝐵𝑥𝐴) → ((1st ↾ (V × {𝐶}))‘⟨𝑥, 𝐶⟩) = 𝑥)
70 f1ocnvfv 6726 . . . . . . . 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 6379 . . . . . 6 ((𝐶𝐵𝑥𝐴) → (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)) = (𝐹‘⟨𝑥, 𝐶⟩))
7372adantll 705 . . . . 5 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)) = (𝐹‘⟨𝑥, 𝐶⟩))
74 df-ov 6845 . . . . 5 (𝑥𝐹𝐶) = (𝐹‘⟨𝑥, 𝐶⟩)
7573, 74syl6eqr 2817 . . . 4 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐹‘((1st ↾ (V × {𝐶}))‘𝑥)) = (𝑥𝐹𝐶))
7654, 75eqtrd 2799 . . 3 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) ∧ 𝑥𝐴) → (𝐺𝑥) = (𝑥𝐹𝐶))
7776mpteq2dva 4903 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → (𝑥𝐴 ↦ (𝐺𝑥)) = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
7845, 77eqtrd 2799 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵) → 𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  cin 3731  wss 3732  {csn 4334  cop 4340  cmpt 4888   × cxp 5275  ccnv 5276  dom cdm 5277  ran crn 5278  cres 5279  cima 5280  ccom 5281  Fun wfun 6062   Fn wfn 6063  ontowfo 6066  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  1st c1st 7364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-1st 7366  df-2nd 7367
This theorem is referenced by:  curry2f  7475  curry2val  7476
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