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Theorem isofn 17709
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-2020.)
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 7881 . . . . . 6 (𝑥 ∈ V → dom 𝑥 ∈ V)
21adantl 483 . . . . 5 ((𝐶 ∈ Cat ∧ 𝑥 ∈ V) → dom 𝑥 ∈ V)
32ralrimiva 3147 . . . 4 (𝐶 ∈ Cat → ∀𝑥 ∈ V dom 𝑥 ∈ V)
4 eqid 2733 . . . . 5 (𝑥 ∈ V ↦ dom 𝑥) = (𝑥 ∈ V ↦ dom 𝑥)
54fnmpt 6680 . . . 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 7429 . . . . . . . 8 (𝑥(Sect‘𝐶)𝑦) ∈ V
87inex1 5313 . . . . . . 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 3201 . . . . 5 (𝐶 ∈ Cat → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)) ∈ V)
11 eqid 2733 . . . . . 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)))
1211fnmpo 8042 . . . . 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 17682 . . . . . 6 Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
15 fveq2 6881 . . . . . . 7 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
16 fveq2 6881 . . . . . . . . 9 (𝑐 = 𝐶 → (Sect‘𝑐) = (Sect‘𝐶))
1716oveqd 7413 . . . . . . . 8 (𝑐 = 𝐶 → (𝑥(Sect‘𝑐)𝑦) = (𝑥(Sect‘𝐶)𝑦))
1816oveqd 7413 . . . . . . . . 9 (𝑐 = 𝐶 → (𝑦(Sect‘𝑐)𝑥) = (𝑦(Sect‘𝐶)𝑥))
1918cnveqd 5870 . . . . . . . 8 (𝑐 = 𝐶(𝑦(Sect‘𝑐)𝑥) = (𝑦(Sect‘𝐶)𝑥))
2017, 19ineq12d 4211 . . . . . . 7 (𝑐 = 𝐶 → ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥)) = ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)))
2115, 15, 20mpoeq123dv 7471 . . . . . 6 (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))))
22 id 22 . . . . . 6 (𝐶 ∈ Cat → 𝐶 ∈ Cat)
23 fvex 6894 . . . . . . . 8 (Base‘𝐶) ∈ V
2423, 23pm3.2i 472 . . . . . . 7 ((Base‘𝐶) ∈ V ∧ (Base‘𝐶) ∈ V)
25 mpoexga 8051 . . . . . . 7 (((Base‘𝐶) ∈ V ∧ (Base‘𝐶) ∈ V) → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) ∈ V)
2624, 25mp1i 13 . . . . . 6 (𝐶 ∈ Cat → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) ∈ V)
2714, 21, 22, 26fvmptd3 7010 . . . . 5 (𝐶 ∈ Cat → (Inv‘𝐶) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))))
2827fneq1d 6634 . . . 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))))
2913, 28mpbird 257 . . 3 (𝐶 ∈ Cat → (Inv‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
30 ssv 4004 . . . 4 ran (Inv‘𝐶) ⊆ V
3130a1i 11 . . 3 (𝐶 ∈ Cat → ran (Inv‘𝐶) ⊆ V)
32 fnco 6657 . . 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)))
336, 29, 31, 32syl3anc 1372 . 2 (𝐶 ∈ Cat → ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶)))
34 isofval 17691 . . 3 (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)))
3534fneq1d 6634 . 2 (𝐶 ∈ Cat → ((Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝐶)) Fn ((Base‘𝐶) × (Base‘𝐶))))
3633, 35mpbird 257 1 (𝐶 ∈ Cat → (Iso‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cin 3945  wss 3946  cmpt 5227   × cxp 5670  ccnv 5671  dom cdm 5672  ran crn 5673  ccom 5676   Fn wfn 6530  cfv 6535  (class class class)co 7396  cmpo 7398  Basecbs 17131  Catccat 17595  Sectcsect 17678  Invcinv 17679  Isociso 17680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7962  df-2nd 7963  df-inv 17682  df-iso 17683
This theorem is referenced by:  brcic  17732  ciclcl  17736  cicrcl  17737  cicer  17740
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