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Theorem isfuncd 16572
Description: Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
isfunc.b 𝐵 = (Base‘𝐷)
isfunc.c 𝐶 = (Base‘𝐸)
isfunc.h 𝐻 = (Hom ‘𝐷)
isfunc.j 𝐽 = (Hom ‘𝐸)
isfunc.1 1 = (Id‘𝐷)
isfunc.i 𝐼 = (Id‘𝐸)
isfunc.x · = (comp‘𝐷)
isfunc.o 𝑂 = (comp‘𝐸)
isfunc.d (𝜑𝐷 ∈ Cat)
isfunc.e (𝜑𝐸 ∈ Cat)
isfuncd.1 (𝜑𝐹:𝐵𝐶)
isfuncd.2 (𝜑𝐺 Fn (𝐵 × 𝐵))
isfuncd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))
isfuncd.4 ((𝜑𝑥𝐵) → ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)))
isfuncd.5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))
Assertion
Ref Expression
isfuncd (𝜑𝐹(𝐷 Func 𝐸)𝐺)
Distinct variable groups:   𝑚,𝑛,𝑥,𝑦,𝑧,𝐵   𝐷,𝑚,𝑛,𝑥,𝑦,𝑧   𝑚,𝐸,𝑛,𝑥,𝑦,𝑧   𝑚,𝐻,𝑛,𝑥,𝑦,𝑧   𝑚,𝐹,𝑛,𝑥,𝑦,𝑧   𝑚,𝐺,𝑛,𝑥,𝑦,𝑧   𝑥,𝐽,𝑦,𝑧   𝜑,𝑚,𝑛,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑚,𝑛)   · (𝑥,𝑦,𝑧,𝑚,𝑛)   1 (𝑥,𝑦,𝑧,𝑚,𝑛)   𝐼(𝑥,𝑦,𝑧,𝑚,𝑛)   𝐽(𝑚,𝑛)   𝑂(𝑥,𝑦,𝑧,𝑚,𝑛)

Proof of Theorem isfuncd
StepHypRef Expression
1 isfuncd.1 . 2 (𝜑𝐹:𝐵𝐶)
2 isfuncd.2 . . . 4 (𝜑𝐺 Fn (𝐵 × 𝐵))
3 isfunc.b . . . . . 6 𝐵 = (Base‘𝐷)
4 fvex 6239 . . . . . 6 (Base‘𝐷) ∈ V
53, 4eqeltri 2726 . . . . 5 𝐵 ∈ V
65, 5xpex 7004 . . . 4 (𝐵 × 𝐵) ∈ V
7 fnex 6522 . . . 4 ((𝐺 Fn (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ∈ V) → 𝐺 ∈ V)
82, 6, 7sylancl 695 . . 3 (𝜑𝐺 ∈ V)
9 isfuncd.3 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))
10 ovex 6718 . . . . . . 7 ((𝐹𝑥)𝐽(𝐹𝑦)) ∈ V
11 ovex 6718 . . . . . . 7 (𝑥𝐻𝑦) ∈ V
1210, 11elmap 7928 . . . . . 6 ((𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))
139, 12sylibr 224 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)))
1413ralrimivva 3000 . . . 4 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)))
15 fveq2 6229 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
16 df-ov 6693 . . . . . . 7 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
1715, 16syl6eqr 2703 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
18 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
19 vex 3234 . . . . . . . . . 10 𝑦 ∈ V
2018, 19op1std 7220 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
2120fveq2d 6233 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑧)) = (𝐹𝑥))
2218, 19op2ndd 7221 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
2322fveq2d 6233 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd𝑧)) = (𝐹𝑦))
2421, 23oveq12d 6708 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) = ((𝐹𝑥)𝐽(𝐹𝑦)))
25 fveq2 6229 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
26 df-ov 6693 . . . . . . . 8 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
2725, 26syl6eqr 2703 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
2824, 27oveq12d 6708 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) = (((𝐹𝑥)𝐽(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)))
2917, 28eleq12d 2724 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦))))
3029ralxp 5296 . . . 4 (∀𝑧 ∈ (𝐵 × 𝐵)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑𝑚 (𝑥𝐻𝑦)))
3114, 30sylibr 224 . . 3 (𝜑 → ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))
32 elixp2 7954 . . 3 (𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧))))
338, 2, 31, 32syl3anbrc 1265 . 2 (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))
34 isfuncd.4 . . . 4 ((𝜑𝑥𝐵) → ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)))
35 isfuncd.5 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))
36353expia 1286 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
37363exp2 1307 . . . . . . 7 (𝜑 → (𝑥𝐵 → (𝑦𝐵 → (𝑧𝐵 → ((𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))))))
3837imp43 620 . . . . . 6 (((𝜑𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → ((𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
3938ralrimivv 2999 . . . . 5 (((𝜑𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))
4039ralrimivva 3000 . . . 4 ((𝜑𝑥𝐵) → ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))
4134, 40jca 553 . . 3 ((𝜑𝑥𝐵) → (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
4241ralrimiva 2995 . 2 (𝜑 → ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
43 isfunc.c . . 3 𝐶 = (Base‘𝐸)
44 isfunc.h . . 3 𝐻 = (Hom ‘𝐷)
45 isfunc.j . . 3 𝐽 = (Hom ‘𝐸)
46 isfunc.1 . . 3 1 = (Id‘𝐷)
47 isfunc.i . . 3 𝐼 = (Id‘𝐸)
48 isfunc.x . . 3 · = (comp‘𝐷)
49 isfunc.o . . 3 𝑂 = (comp‘𝐸)
50 isfunc.d . . 3 (𝜑𝐷 ∈ Cat)
51 isfunc.e . . 3 (𝜑𝐸 ∈ Cat)
523, 43, 44, 45, 46, 47, 48, 49, 50, 51isfunc 16571 . 2 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵𝐶𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))
531, 33, 42, 52mpbir3and 1264 1 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cop 4216   class class class wbr 4685   × cxp 5141   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Xcixp 7950  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373   Func cfunc 16561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-ixp 7951  df-func 16565
This theorem is referenced by:  funcoppc  16582  funcres  16603  catcisolem  16803  funcestrcsetc  16836  funcsetcestrc  16851  1stfcl  16884  2ndfcl  16885  prfcl  16890  evlfcl  16909  curf1cl  16915  curfcl  16919  hofcl  16946  funcringcsetcALTV2  42370  funcringcsetcALTV  42393
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