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Theorem isfuncd 17127
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)
43fvexi 6659 . . . . 5 𝐵 ∈ V
54, 4xpex 7456 . . . 4 (𝐵 × 𝐵) ∈ V
6 fnex 6957 . . . 4 ((𝐺 Fn (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ∈ V) → 𝐺 ∈ V)
72, 5, 6sylancl 589 . . 3 (𝜑𝐺 ∈ V)
8 isfuncd.3 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))
9 ovex 7168 . . . . . . 7 ((𝐹𝑥)𝐽(𝐹𝑦)) ∈ V
10 ovex 7168 . . . . . . 7 (𝑥𝐻𝑦) ∈ V
119, 10elmap 8418 . . . . . 6 ((𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑m (𝑥𝐻𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹𝑥)𝐽(𝐹𝑦)))
128, 11sylibr 237 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑m (𝑥𝐻𝑦)))
1312ralrimivva 3156 . . . 4 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑m (𝑥𝐻𝑦)))
14 fveq2 6645 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
15 df-ov 7138 . . . . . . 7 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
1614, 15eqtr4di 2851 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
17 vex 3444 . . . . . . . . . 10 𝑥 ∈ V
18 vex 3444 . . . . . . . . . 10 𝑦 ∈ V
1917, 18op1std 7681 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
2019fveq2d 6649 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑧)) = (𝐹𝑥))
2117, 18op2ndd 7682 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
2221fveq2d 6649 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd𝑧)) = (𝐹𝑦))
2320, 22oveq12d 7153 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) = ((𝐹𝑥)𝐽(𝐹𝑦)))
24 fveq2 6645 . . . . . . . 8 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
25 df-ov 7138 . . . . . . . 8 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
2624, 25eqtr4di 2851 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
2723, 26oveq12d 7153 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) = (((𝐹𝑥)𝐽(𝐹𝑦)) ↑m (𝑥𝐻𝑦)))
2816, 27eleq12d 2884 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ (𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑m (𝑥𝐻𝑦))))
2928ralxp 5676 . . . 4 (∀𝑧 ∈ (𝐵 × 𝐵)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝐺𝑦) ∈ (((𝐹𝑥)𝐽(𝐹𝑦)) ↑m (𝑥𝐻𝑦)))
3013, 29sylibr 237 . . 3 (𝜑 → ∀𝑧 ∈ (𝐵 × 𝐵)(𝐺𝑧) ∈ (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)))
31 elixp2 8448 . . 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))))
327, 2, 30, 31syl3anbrc 1340 . 2 (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)))
33 isfuncd.4 . . . 4 ((𝜑𝑥𝐵) → ((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)))
34 isfuncd.5 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵) ∧ (𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧))) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))
35343expia 1118 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
36353exp2 1351 . . . . . . 7 (𝜑 → (𝑥𝐵 → (𝑦𝐵 → (𝑧𝐵 → ((𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))))))
3736imp43 431 . . . . . 6 (((𝜑𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → ((𝑚 ∈ (𝑥𝐻𝑦) ∧ 𝑛 ∈ (𝑦𝐻𝑧)) → ((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
3837ralrimivv 3155 . . . . 5 (((𝜑𝑥𝐵) ∧ (𝑦𝐵𝑧𝐵)) → ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))
3938ralrimivva 3156 . . . 4 ((𝜑𝑥𝐵) → ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))
4033, 39jca 515 . . 3 ((𝜑𝑥𝐵) → (((𝑥𝐺𝑥)‘( 1𝑥)) = (𝐼‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦· 𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩𝑂(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))
4140ralrimiva 3149 . 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))))
42 isfunc.c . . 3 𝐶 = (Base‘𝐸)
43 isfunc.h . . 3 𝐻 = (Hom ‘𝐷)
44 isfunc.j . . 3 𝐽 = (Hom ‘𝐸)
45 isfunc.1 . . 3 1 = (Id‘𝐷)
46 isfunc.i . . 3 𝐼 = (Id‘𝐸)
47 isfunc.x . . 3 · = (comp‘𝐷)
48 isfunc.o . . 3 𝑂 = (comp‘𝐸)
49 isfunc.d . . 3 (𝜑𝐷 ∈ Cat)
50 isfunc.e . . 3 (𝜑𝐸 ∈ Cat)
513, 42, 43, 44, 45, 46, 47, 48, 49, 50isfunc 17126 . 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))))))
521, 32, 41, 51mpbir3and 1339 1 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  Vcvv 3441  cop 4531   class class class wbr 5030   × cxp 5517   Fn wfn 6319  wf 6320  cfv 6324  (class class class)co 7135  1st c1st 7669  2nd c2nd 7670  m cmap 8389  Xcixp 8444  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Idccid 16928   Func cfunc 17116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-ixp 8445  df-func 17120
This theorem is referenced by:  funcoppc  17137  funcres  17158  catcisolem  17358  funcestrcsetc  17391  funcsetcestrc  17406  1stfcl  17439  2ndfcl  17440  prfcl  17445  evlfcl  17464  curf1cl  17470  curfcl  17474  hofcl  17501  funcringcsetcALTV2  44664  funcringcsetcALTV  44687
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