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Theorem isoval 16632
Description: The isomorphisms are the domain of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 21-May-2020.)
Hypotheses
Ref Expression
invfval.b 𝐵 = (Base‘𝐶)
invfval.n 𝑁 = (Inv‘𝐶)
invfval.c (𝜑𝐶 ∈ Cat)
invfval.x (𝜑𝑋𝐵)
invfval.y (𝜑𝑌𝐵)
isoval.n 𝐼 = (Iso‘𝐶)
Assertion
Ref Expression
isoval (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))

Proof of Theorem isoval
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfval.c . . . . 5 (𝜑𝐶 ∈ Cat)
2 isofval 16624 . . . . 5 (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶)))
31, 2syl 17 . . . 4 (𝜑 → (Iso‘𝐶) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶)))
4 isoval.n . . . 4 𝐼 = (Iso‘𝐶)
5 invfval.n . . . . 5 𝑁 = (Inv‘𝐶)
65coeq2i 5420 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶))
73, 4, 63eqtr4g 2830 . . 3 (𝜑𝐼 = ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁))
87oveqd 6813 . 2 (𝜑 → (𝑋𝐼𝑌) = (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌))
9 eqid 2771 . . . . . 6 (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)))
10 ovex 6827 . . . . . . 7 (𝑥(Sect‘𝐶)𝑦) ∈ V
1110inex1 4934 . . . . . 6 ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)) ∈ V
129, 11fnmpt2i 7393 . . . . 5 (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) Fn (𝐵 × 𝐵)
13 invfval.b . . . . . . 7 𝐵 = (Base‘𝐶)
14 invfval.x . . . . . . 7 (𝜑𝑋𝐵)
15 invfval.y . . . . . . 7 (𝜑𝑌𝐵)
16 eqid 2771 . . . . . . 7 (Sect‘𝐶) = (Sect‘𝐶)
1713, 5, 1, 14, 15, 16invffval 16625 . . . . . 6 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))))
1817fneq1d 6120 . . . . 5 (𝜑 → (𝑁 Fn (𝐵 × 𝐵) ↔ (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) Fn (𝐵 × 𝐵)))
1912, 18mpbiri 248 . . . 4 (𝜑𝑁 Fn (𝐵 × 𝐵))
20 opelxpi 5287 . . . . 5 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
2114, 15, 20syl2anc 573 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
22 fvco2 6417 . . . 4 ((𝑁 Fn (𝐵 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵)) → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
2319, 21, 22syl2anc 573 . . 3 (𝜑 → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
24 df-ov 6799 . . 3 (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩)
25 ovex 6827 . . . . 5 (𝑋𝑁𝑌) ∈ V
26 dmeq 5461 . . . . . 6 (𝑧 = (𝑋𝑁𝑌) → dom 𝑧 = dom (𝑋𝑁𝑌))
27 eqid 2771 . . . . . 6 (𝑧 ∈ V ↦ dom 𝑧) = (𝑧 ∈ V ↦ dom 𝑧)
2825dmex 7250 . . . . . 6 dom (𝑋𝑁𝑌) ∈ V
2926, 27, 28fvmpt 6426 . . . . 5 ((𝑋𝑁𝑌) ∈ V → ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = dom (𝑋𝑁𝑌))
3025, 29ax-mp 5 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = dom (𝑋𝑁𝑌)
31 df-ov 6799 . . . . 5 (𝑋𝑁𝑌) = (𝑁‘⟨𝑋, 𝑌⟩)
3231fveq2i 6336 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
3330, 32eqtr3i 2795 . . 3 dom (𝑋𝑁𝑌) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
3423, 24, 333eqtr4g 2830 . 2 (𝜑 → (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = dom (𝑋𝑁𝑌))
358, 34eqtrd 2805 1 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cin 3722  cop 4323  cmpt 4864   × cxp 5248  ccnv 5249  dom cdm 5250  ccom 5254   Fn wfn 6025  cfv 6030  (class class class)co 6796  cmpt2 6798  Basecbs 16064  Catccat 16532  Sectcsect 16611  Invcinv 16612  Isociso 16613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-inv 16615  df-iso 16616
This theorem is referenced by:  inviso1  16633  invf  16635  invco  16638  dfiso2  16639  isohom  16643  oppciso  16648  cicsym  16671  funciso  16741  ffthiso  16796  fuciso  16842  setciso  16948  catciso  16964  rngciso  42507  rngcisoALTV  42519  ringciso  42558  ringcisoALTV  42582
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