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Theorem invsym2 17739
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 2728 . . . . 5 (Hom β€˜πΆ) = (Hom β€˜πΆ)
71, 2, 3, 4, 5, 6invss 17737 . . . 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 5690 . . . 4 Rel ((π‘Œ(Hom β€˜πΆ)𝑋) Γ— (𝑋(Hom β€˜πΆ)π‘Œ))
9 relss 5777 . . . 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 6102 . . 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 17738 . . . 4 (πœ‘ β†’ (𝑓(π‘‹π‘π‘Œ)𝑔 ↔ 𝑔(π‘Œπ‘π‘‹)𝑓))
14 vex 3474 . . . . . 6 𝑔 ∈ V
15 vex 3474 . . . . . 6 𝑓 ∈ V
1614, 15brcnv 5879 . . . . 5 (𝑔◑(π‘‹π‘π‘Œ)𝑓 ↔ 𝑓(π‘‹π‘π‘Œ)𝑔)
17 df-br 5143 . . . . 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 5143 . . . 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 5791 . 2 (((Rel β—‘(π‘‹π‘π‘Œ) ∧ Rel (π‘Œπ‘π‘‹)) ∧ πœ‘) β†’ β—‘(π‘‹π‘π‘Œ) = (π‘Œπ‘π‘‹))
2212, 21mpancom 687 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 1534   ∈ wcel 2099   βŠ† wss 3945  βŸ¨cop 4630   class class class wbr 5142   Γ— cxp 5670  β—‘ccnv 5671  Rel wrel 5677  β€˜cfv 6542  (class class class)co 7414  Basecbs 17173  Hom chom 17237  Catccat 17637  Invcinv 17721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-sect 17723  df-inv 17724
This theorem is referenced by:  invf  17744  invf1o  17745  invinv  17746  cicsym  17780
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