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Theorem isofn 16416
Description: The function value of the function returning the isomorphisms of a category is a function over the square product of the base set of the category. (Contributed by AV, 5-Apr-2017.)
Assertion
Ref Expression
isofn (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))

Proof of Theorem isofn
Dummy variables 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmexg 7082 . . . . . 6 (𝑥 ∈ V → dom 𝑥 ∈ V)
21adantl 482 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑥 ∈ V) → dom 𝑥 ∈ V)
32ralrimiva 2963 . . . 4 (𝐶 ∈ Cat → ∀𝑥 ∈ V dom 𝑥 ∈ V)
4 eqid 2620 . . . . 5 (𝑥 ∈ V ↦ dom 𝑥) = (𝑥 ∈ V ↦ dom 𝑥)
54fnmpt 6007 . . . 4 (∀𝑥 ∈ V dom 𝑥 ∈ V → (𝑥 ∈ V ↦ dom 𝑥) Fn V)
63, 5syl 17 . . 3 (𝐶 ∈ Cat → (𝑥 ∈ V ↦ dom 𝑥) Fn V)
7 ovex 6663 . . . . . . . 8 (𝑥(Sect‘𝐶)𝑦) ∈ V
87inex1 4790 . . . . . . 7 ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)) ∈ V
98a1i 11 . . . . . 6 ((𝐶 ∈ Cat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)) ∈ V)
109ralrimivva 2968 . . . . 5 (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)) ∈ V)
11 eqid 2620 . . . . . 6 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)))
1211fnmpt2 7223 . . . . 5 (∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)) ∈ V → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) Fn ((Base‘𝐶) × (Base‘𝐶)))
1310, 12syl 17 . . . 4 (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) Fn ((Base‘𝐶) × (Base‘𝐶)))
14 df-inv 16389 . . . . . . 7 Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
1514a1i 11 . . . . . 6 (𝐶 ∈ Cat → Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥)))))
16 fveq2 6178 . . . . . . . 8 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
17 fveq2 6178 . . . . . . . . . 10 (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
1817oveqd 6652 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥(Sect‘𝐶)𝑦))
1917oveqd 6652 . . . . . . . . . 10 (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦(Sect‘𝐶)𝑥))
2019cnveqd 5287 . . . . . . . . 9 (𝑐 = 𝐶(𝑦(Sect‘𝑐)𝑥) = (𝑦(Sect‘𝐶)𝑥))
2118, 20ineq12d 3807 . . . . . . . 8 (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥)) = ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)))
2216, 16, 21mpt2eq123dv 6702 . . . . . . 7 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))))
2322adantl 482 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝑐 = 𝐶) → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))))
24 id 22 . . . . . 6 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
25 fvex 6188 . . . . . . . 8 (Base‘𝐶) ∈ V
2625, 25pm3.2i 471 . . . . . . 7 ((Base‘𝐶) ∈ V ∧ (Base‘𝐶) ∈ V)
2711mpt2exg 7230 . . . . . . 7 (((Base‘𝐶) ∈ V ∧ (Base‘𝐶) ∈ V) → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) ∈ V)
2826, 27mp1i 13 . . . . . 6 (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) ∈ V)
2915, 23, 24, 28fvmptd 6275 . . . . 5 (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))))
3029fneq1d 5969 . . . 4 (𝐶 ∈ Cat → ((Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) Fn ((Base‘𝐶) × (Base‘𝐶))))
3113, 30mpbird 247 . . 3 (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
32 ssv 3617 . . . 4 ran (Inv‘𝐶) ⊆ V
3332a1i 11 . . 3 (𝐶 ∈ Cat → ran (Inv‘𝐶) ⊆ V)
34 fnco 5987 . . 3 (((𝑥 ∈ V ↦ dom 𝑥) Fn V ∧ (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ∧ ran (Inv‘𝐶) ⊆ V) → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶)))
356, 31, 33, 34syl3anc 1324 . 2 (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶)))
36 isofval 16398 . . 3 (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
3736fneq1d 5969 . 2 (𝐶 ∈ Cat → ((Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶))))
3835, 37mpbird 247 1 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  cin 3566  wss 3567  cmpt 4720   × cxp 5102  ccnv 5103  dom cdm 5104  ran crn 5105  ccom 5108   Fn wfn 5871  cfv 5876  (class class class)co 6635  cmpt2 6637  Basecbs 15838  Catccat 16306  Sectcsect 16385  Invcinv 16386  Isociso 16387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-inv 16389  df-iso 16390
This theorem is referenced by:  brcic  16439  ciclcl  16443  cicrcl  16444  cicer  16447
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