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Theorem 2termoinv 17979
Description: Morphisms between two terminal objects are inverses. (Contributed by AV, 18-Apr-2020.)
Hypotheses
Ref Expression
termoeu1.c (πœ‘ β†’ 𝐢 ∈ Cat)
termoeu1.a (πœ‘ β†’ 𝐴 ∈ (TermOβ€˜πΆ))
termoeu1.b (πœ‘ β†’ 𝐡 ∈ (TermOβ€˜πΆ))
Assertion
Ref Expression
2termoinv ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐹(𝐴(Invβ€˜πΆ)𝐡)𝐺)

Proof of Theorem 2termoinv
StepHypRef Expression
1 eqid 2726 . . . . 5 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
2 eqid 2726 . . . . 5 (Hom β€˜πΆ) = (Hom β€˜πΆ)
3 eqid 2726 . . . . 5 (compβ€˜πΆ) = (compβ€˜πΆ)
4 termoeu1.c . . . . . 6 (πœ‘ β†’ 𝐢 ∈ Cat)
543ad2ant1 1130 . . . . 5 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐢 ∈ Cat)
6 termoeu1.a . . . . . . 7 (πœ‘ β†’ 𝐴 ∈ (TermOβ€˜πΆ))
7 termoo 17970 . . . . . . 7 (𝐢 ∈ Cat β†’ (𝐴 ∈ (TermOβ€˜πΆ) β†’ 𝐴 ∈ (Baseβ€˜πΆ)))
84, 6, 7sylc 65 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ (Baseβ€˜πΆ))
983ad2ant1 1130 . . . . 5 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐴 ∈ (Baseβ€˜πΆ))
10 termoeu1.b . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ (TermOβ€˜πΆ))
11 termoo 17970 . . . . . . 7 (𝐢 ∈ Cat β†’ (𝐡 ∈ (TermOβ€˜πΆ) β†’ 𝐡 ∈ (Baseβ€˜πΆ)))
124, 10, 11sylc 65 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ (Baseβ€˜πΆ))
13123ad2ant1 1130 . . . . 5 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐡 ∈ (Baseβ€˜πΆ))
14 simp3 1135 . . . . 5 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡))
15 simp2 1134 . . . . 5 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴))
161, 2, 3, 5, 9, 13, 9, 14, 15catcocl 17638 . . . 4 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ (𝐺(⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐴)𝐹) ∈ (𝐴(Hom β€˜πΆ)𝐴))
171, 2, 4termoid 17964 . . . . . . . 8 ((πœ‘ ∧ 𝐴 ∈ (TermOβ€˜πΆ)) β†’ (𝐴(Hom β€˜πΆ)𝐴) = {((Idβ€˜πΆ)β€˜π΄)})
186, 17mpdan 684 . . . . . . 7 (πœ‘ β†’ (𝐴(Hom β€˜πΆ)𝐴) = {((Idβ€˜πΆ)β€˜π΄)})
19183ad2ant1 1130 . . . . . 6 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ (𝐴(Hom β€˜πΆ)𝐴) = {((Idβ€˜πΆ)β€˜π΄)})
2019eleq2d 2813 . . . . 5 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ ((𝐺(⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐴)𝐹) ∈ (𝐴(Hom β€˜πΆ)𝐴) ↔ (𝐺(⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐴)𝐹) ∈ {((Idβ€˜πΆ)β€˜π΄)}))
21 elsni 4640 . . . . 5 ((𝐺(⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐴)𝐹) ∈ {((Idβ€˜πΆ)β€˜π΄)} β†’ (𝐺(⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐴)𝐹) = ((Idβ€˜πΆ)β€˜π΄))
2220, 21biimtrdi 252 . . . 4 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ ((𝐺(⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐴)𝐹) ∈ (𝐴(Hom β€˜πΆ)𝐴) β†’ (𝐺(⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐴)𝐹) = ((Idβ€˜πΆ)β€˜π΄)))
2316, 22mpd 15 . . 3 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ (𝐺(⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐴)𝐹) = ((Idβ€˜πΆ)β€˜π΄))
24 eqid 2726 . . . 4 (Idβ€˜πΆ) = (Idβ€˜πΆ)
25 eqid 2726 . . . 4 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
261, 2, 3, 24, 25, 5, 9, 13, 14, 15issect2 17710 . . 3 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ (𝐹(𝐴(Sectβ€˜πΆ)𝐡)𝐺 ↔ (𝐺(⟨𝐴, 𝐡⟩(compβ€˜πΆ)𝐴)𝐹) = ((Idβ€˜πΆ)β€˜π΄)))
2723, 26mpbird 257 . 2 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐹(𝐴(Sectβ€˜πΆ)𝐡)𝐺)
281, 2, 3, 5, 13, 9, 13, 15, 14catcocl 17638 . . . 4 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ (𝐹(⟨𝐡, 𝐴⟩(compβ€˜πΆ)𝐡)𝐺) ∈ (𝐡(Hom β€˜πΆ)𝐡))
291, 2, 4termoid 17964 . . . . . . . 8 ((πœ‘ ∧ 𝐡 ∈ (TermOβ€˜πΆ)) β†’ (𝐡(Hom β€˜πΆ)𝐡) = {((Idβ€˜πΆ)β€˜π΅)})
3010, 29mpdan 684 . . . . . . 7 (πœ‘ β†’ (𝐡(Hom β€˜πΆ)𝐡) = {((Idβ€˜πΆ)β€˜π΅)})
31303ad2ant1 1130 . . . . . 6 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ (𝐡(Hom β€˜πΆ)𝐡) = {((Idβ€˜πΆ)β€˜π΅)})
3231eleq2d 2813 . . . . 5 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ ((𝐹(⟨𝐡, 𝐴⟩(compβ€˜πΆ)𝐡)𝐺) ∈ (𝐡(Hom β€˜πΆ)𝐡) ↔ (𝐹(⟨𝐡, 𝐴⟩(compβ€˜πΆ)𝐡)𝐺) ∈ {((Idβ€˜πΆ)β€˜π΅)}))
33 elsni 4640 . . . . 5 ((𝐹(⟨𝐡, 𝐴⟩(compβ€˜πΆ)𝐡)𝐺) ∈ {((Idβ€˜πΆ)β€˜π΅)} β†’ (𝐹(⟨𝐡, 𝐴⟩(compβ€˜πΆ)𝐡)𝐺) = ((Idβ€˜πΆ)β€˜π΅))
3432, 33biimtrdi 252 . . . 4 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ ((𝐹(⟨𝐡, 𝐴⟩(compβ€˜πΆ)𝐡)𝐺) ∈ (𝐡(Hom β€˜πΆ)𝐡) β†’ (𝐹(⟨𝐡, 𝐴⟩(compβ€˜πΆ)𝐡)𝐺) = ((Idβ€˜πΆ)β€˜π΅)))
3528, 34mpd 15 . . 3 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ (𝐹(⟨𝐡, 𝐴⟩(compβ€˜πΆ)𝐡)𝐺) = ((Idβ€˜πΆ)β€˜π΅))
361, 2, 3, 24, 25, 5, 13, 9, 15, 14issect2 17710 . . 3 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ (𝐺(𝐡(Sectβ€˜πΆ)𝐴)𝐹 ↔ (𝐹(⟨𝐡, 𝐴⟩(compβ€˜πΆ)𝐡)𝐺) = ((Idβ€˜πΆ)β€˜π΅)))
3735, 36mpbird 257 . 2 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐺(𝐡(Sectβ€˜πΆ)𝐴)𝐹)
38 eqid 2726 . . . 4 (Invβ€˜πΆ) = (Invβ€˜πΆ)
391, 38, 4, 8, 12, 25isinv 17716 . . 3 (πœ‘ β†’ (𝐹(𝐴(Invβ€˜πΆ)𝐡)𝐺 ↔ (𝐹(𝐴(Sectβ€˜πΆ)𝐡)𝐺 ∧ 𝐺(𝐡(Sectβ€˜πΆ)𝐴)𝐹)))
40393ad2ant1 1130 . 2 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ (𝐹(𝐴(Invβ€˜πΆ)𝐡)𝐺 ↔ (𝐹(𝐴(Sectβ€˜πΆ)𝐡)𝐺 ∧ 𝐺(𝐡(Sectβ€˜πΆ)𝐴)𝐹)))
4127, 37, 40mpbir2and 710 1 ((πœ‘ ∧ 𝐺 ∈ (𝐡(Hom β€˜πΆ)𝐴) ∧ 𝐹 ∈ (𝐴(Hom β€˜πΆ)𝐡)) β†’ 𝐹(𝐴(Invβ€˜πΆ)𝐡)𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {csn 4623  βŸ¨cop 4629   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  Hom chom 17217  compcco 17218  Catccat 17617  Idccid 17618  Sectcsect 17700  Invcinv 17701  TermOctermo 17944
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-cat 17621  df-cid 17622  df-sect 17703  df-inv 17704  df-termo 17947
This theorem is referenced by:  termoeu1  17980
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