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Theorem cicsym 17065
Description: Isomorphism is symmetric. (Contributed by AV, 5-Apr-2020.)
Assertion
Ref Expression
cicsym ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆( ≃𝑐𝐶)𝑅)

Proof of Theorem cicsym
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cicrcl 17064 . 2 ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆 ∈ (Base‘𝐶))
2 ciclcl 17063 . 2 ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑅 ∈ (Base‘𝐶))
3 eqid 2822 . . . . 5 (Iso‘𝐶) = (Iso‘𝐶)
4 eqid 2822 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
5 simpl 486 . . . . 5 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝐶 ∈ Cat)
6 simpr 488 . . . . . 6 ((𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶)) → 𝑅 ∈ (Base‘𝐶))
76adantl 485 . . . . 5 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑅 ∈ (Base‘𝐶))
8 simpl 486 . . . . . 6 ((𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶)) → 𝑆 ∈ (Base‘𝐶))
98adantl 485 . . . . 5 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆 ∈ (Base‘𝐶))
103, 4, 5, 7, 9cic 17060 . . . 4 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐𝐶)𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆)))
11 eqid 2822 . . . . . . . . . 10 (Inv‘𝐶) = (Inv‘𝐶)
124, 11, 5, 7, 9, 3isoval 17026 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Iso‘𝐶)𝑆) = dom (𝑅(Inv‘𝐶)𝑆))
134, 11, 5, 9, 7invsym2 17024 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑆(Inv‘𝐶)𝑅) = (𝑅(Inv‘𝐶)𝑆))
1413eqcomd 2828 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Inv‘𝐶)𝑆) = (𝑆(Inv‘𝐶)𝑅))
1514dmeqd 5751 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑅(Inv‘𝐶)𝑆) = dom (𝑆(Inv‘𝐶)𝑅))
16 df-rn 5543 . . . . . . . . . 10 ran (𝑆(Inv‘𝐶)𝑅) = dom (𝑆(Inv‘𝐶)𝑅)
1715, 16eqtr4di 2875 . . . . . . . . 9 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑅(Inv‘𝐶)𝑆) = ran (𝑆(Inv‘𝐶)𝑅))
1812, 17eqtrd 2857 . . . . . . . 8 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅(Iso‘𝐶)𝑆) = ran (𝑆(Inv‘𝐶)𝑅))
1918eleq2d 2899 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ↔ 𝑓 ∈ ran (𝑆(Inv‘𝐶)𝑅)))
20 vex 3472 . . . . . . . 8 𝑓 ∈ V
21 elrng 5739 . . . . . . . 8 (𝑓 ∈ V → (𝑓 ∈ ran (𝑆(Inv‘𝐶)𝑅) ↔ ∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓))
2220, 21mp1i 13 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ ran (𝑆(Inv‘𝐶)𝑅) ↔ ∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓))
2319, 22bitrd 282 . . . . . 6 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) ↔ ∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓))
24 df-br 5043 . . . . . . . 8 (𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 ↔ ⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅))
2524exbii 1849 . . . . . . 7 (∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓 ↔ ∃𝑔𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅))
26 vex 3472 . . . . . . . . . . 11 𝑔 ∈ V
2726, 20opeldm 5753 . . . . . . . . . 10 (⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → 𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅))
284, 11, 5, 9, 7, 3isoval 17026 . . . . . . . . . . . . 13 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑆(Iso‘𝐶)𝑅) = dom (𝑆(Inv‘𝐶)𝑅))
2928eqcomd 2828 . . . . . . . . . . . 12 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → dom (𝑆(Inv‘𝐶)𝑅) = (𝑆(Iso‘𝐶)𝑅))
3029eleq2d 2899 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅) ↔ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)))
315adantr 484 . . . . . . . . . . . . 13 (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝐶 ∈ Cat)
329adantr 484 . . . . . . . . . . . . 13 (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑆 ∈ (Base‘𝐶))
337adantr 484 . . . . . . . . . . . . 13 (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑅 ∈ (Base‘𝐶))
34 simpr 488 . . . . . . . . . . . . 13 (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅))
353, 4, 31, 32, 33, 34brcici 17061 . . . . . . . . . . . 12 (((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) ∧ 𝑔 ∈ (𝑆(Iso‘𝐶)𝑅)) → 𝑆( ≃𝑐𝐶)𝑅)
3635ex 416 . . . . . . . . . . 11 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑔 ∈ (𝑆(Iso‘𝐶)𝑅) → 𝑆( ≃𝑐𝐶)𝑅))
3730, 36sylbid 243 . . . . . . . . . 10 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑔 ∈ dom (𝑆(Inv‘𝐶)𝑅) → 𝑆( ≃𝑐𝐶)𝑅))
3827, 37syl5com 31 . . . . . . . . 9 (⟨𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆( ≃𝑐𝐶)𝑅))
3938exlimiv 1931 . . . . . . . 8 (∃𝑔𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → 𝑆( ≃𝑐𝐶)𝑅))
4039com12 32 . . . . . . 7 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑔𝑔, 𝑓⟩ ∈ (𝑆(Inv‘𝐶)𝑅) → 𝑆( ≃𝑐𝐶)𝑅))
4125, 40syl5bi 245 . . . . . 6 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑔 𝑔(𝑆(Inv‘𝐶)𝑅)𝑓𝑆( ≃𝑐𝐶)𝑅))
4223, 41sylbid 243 . . . . 5 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → 𝑆( ≃𝑐𝐶)𝑅))
4342exlimdv 1934 . . . 4 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (∃𝑓 𝑓 ∈ (𝑅(Iso‘𝐶)𝑆) → 𝑆( ≃𝑐𝐶)𝑅))
4410, 43sylbid 243 . . 3 ((𝐶 ∈ Cat ∧ (𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶))) → (𝑅( ≃𝑐𝐶)𝑆𝑆( ≃𝑐𝐶)𝑅))
4544impancom 455 . 2 ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → ((𝑆 ∈ (Base‘𝐶) ∧ 𝑅 ∈ (Base‘𝐶)) → 𝑆( ≃𝑐𝐶)𝑅))
461, 2, 45mp2and 698 1 ((𝐶 ∈ Cat ∧ 𝑅( ≃𝑐𝐶)𝑆) → 𝑆( ≃𝑐𝐶)𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781  wcel 2114  Vcvv 3469  cop 4545   class class class wbr 5042  ccnv 5531  dom cdm 5532  ran crn 5533  cfv 6334  (class class class)co 7140  Basecbs 16474  Catccat 16926  Invcinv 17006  Isociso 17007  𝑐 ccic 17056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-supp 7818  df-sect 17008  df-inv 17009  df-iso 17010  df-cic 17057
This theorem is referenced by:  cicer  17067  initoeu2  17267
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