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Theorem invsym2 17715
Description: The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b 𝐡 = (Baseβ€˜πΆ)
invfval.n 𝑁 = (Invβ€˜πΆ)
invfval.c (πœ‘ β†’ 𝐢 ∈ Cat)
invfval.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
invfval.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
Assertion
Ref Expression
invsym2 (πœ‘ β†’ β—‘(π‘‹π‘π‘Œ) = (π‘Œπ‘π‘‹))

Proof of Theorem invsym2
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfval.b . . . . 5 𝐡 = (Baseβ€˜πΆ)
2 invfval.n . . . . 5 𝑁 = (Invβ€˜πΆ)
3 invfval.c . . . . 5 (πœ‘ β†’ 𝐢 ∈ Cat)
4 invfval.y . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝐡)
5 invfval.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐡)
6 eqid 2724 . . . . 5 (Hom β€˜πΆ) = (Hom β€˜πΆ)
71, 2, 3, 4, 5, 6invss 17713 . . . 4 (πœ‘ β†’ (π‘Œπ‘π‘‹) βŠ† ((π‘Œ(Hom β€˜πΆ)𝑋) Γ— (𝑋(Hom β€˜πΆ)π‘Œ)))
8 relxp 5685 . . . 4 Rel ((π‘Œ(Hom β€˜πΆ)𝑋) Γ— (𝑋(Hom β€˜πΆ)π‘Œ))
9 relss 5772 . . . 4 ((π‘Œπ‘π‘‹) βŠ† ((π‘Œ(Hom β€˜πΆ)𝑋) Γ— (𝑋(Hom β€˜πΆ)π‘Œ)) β†’ (Rel ((π‘Œ(Hom β€˜πΆ)𝑋) Γ— (𝑋(Hom β€˜πΆ)π‘Œ)) β†’ Rel (π‘Œπ‘π‘‹)))
107, 8, 9mpisyl 21 . . 3 (πœ‘ β†’ Rel (π‘Œπ‘π‘‹))
11 relcnv 6094 . . 3 Rel β—‘(π‘‹π‘π‘Œ)
1210, 11jctil 519 . 2 (πœ‘ β†’ (Rel β—‘(π‘‹π‘π‘Œ) ∧ Rel (π‘Œπ‘π‘‹)))
131, 2, 3, 5, 4invsym 17714 . . . 4 (πœ‘ β†’ (𝑓(π‘‹π‘π‘Œ)𝑔 ↔ 𝑔(π‘Œπ‘π‘‹)𝑓))
14 vex 3470 . . . . . 6 𝑔 ∈ V
15 vex 3470 . . . . . 6 𝑓 ∈ V
1614, 15brcnv 5873 . . . . 5 (𝑔◑(π‘‹π‘π‘Œ)𝑓 ↔ 𝑓(π‘‹π‘π‘Œ)𝑔)
17 df-br 5140 . . . . 5 (𝑔◑(π‘‹π‘π‘Œ)𝑓 ↔ βŸ¨π‘”, π‘“βŸ© ∈ β—‘(π‘‹π‘π‘Œ))
1816, 17bitr3i 277 . . . 4 (𝑓(π‘‹π‘π‘Œ)𝑔 ↔ βŸ¨π‘”, π‘“βŸ© ∈ β—‘(π‘‹π‘π‘Œ))
19 df-br 5140 . . . 4 (𝑔(π‘Œπ‘π‘‹)𝑓 ↔ βŸ¨π‘”, π‘“βŸ© ∈ (π‘Œπ‘π‘‹))
2013, 18, 193bitr3g 313 . . 3 (πœ‘ β†’ (βŸ¨π‘”, π‘“βŸ© ∈ β—‘(π‘‹π‘π‘Œ) ↔ βŸ¨π‘”, π‘“βŸ© ∈ (π‘Œπ‘π‘‹)))
2120eqrelrdv2 5786 . 2 (((Rel β—‘(π‘‹π‘π‘Œ) ∧ Rel (π‘Œπ‘π‘‹)) ∧ πœ‘) β†’ β—‘(π‘‹π‘π‘Œ) = (π‘Œπ‘π‘‹))
2212, 21mpancom 685 1 (πœ‘ β†’ β—‘(π‘‹π‘π‘Œ) = (π‘Œπ‘π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βŠ† wss 3941  βŸ¨cop 4627   class class class wbr 5139   Γ— cxp 5665  β—‘ccnv 5666  Rel wrel 5672  β€˜cfv 6534  (class class class)co 7402  Basecbs 17149  Hom chom 17213  Catccat 17613  Invcinv 17697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-sect 17699  df-inv 17700
This theorem is referenced by:  invf  17720  invf1o  17721  invinv  17722  cicsym  17756
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