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Theorem termoid 17998
Description: For a terminal object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 18-Apr-2020.)
Hypotheses
Ref Expression
isinitoi.b 𝐡 = (Baseβ€˜πΆ)
isinitoi.h 𝐻 = (Hom β€˜πΆ)
isinitoi.c (πœ‘ β†’ 𝐢 ∈ Cat)
Assertion
Ref Expression
termoid ((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)})

Proof of Theorem termoid
Dummy variables β„Ž π‘œ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isinitoi.b . . 3 𝐡 = (Baseβ€˜πΆ)
2 isinitoi.h . . 3 𝐻 = (Hom β€˜πΆ)
3 isinitoi.c . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
41, 2, 3istermoi 17996 . 2 ((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) β†’ (𝑂 ∈ 𝐡 ∧ βˆ€π‘œ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘œπ»π‘‚)))
5 oveq1 7433 . . . . . . . 8 (π‘œ = 𝑂 β†’ (π‘œπ»π‘‚) = (𝑂𝐻𝑂))
65eleq2d 2815 . . . . . . 7 (π‘œ = 𝑂 β†’ (β„Ž ∈ (π‘œπ»π‘‚) ↔ β„Ž ∈ (𝑂𝐻𝑂)))
76eubidv 2575 . . . . . 6 (π‘œ = 𝑂 β†’ (βˆƒ!β„Ž β„Ž ∈ (π‘œπ»π‘‚) ↔ βˆƒ!β„Ž β„Ž ∈ (𝑂𝐻𝑂)))
87rspcv 3607 . . . . 5 (𝑂 ∈ 𝐡 β†’ (βˆ€π‘œ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘œπ»π‘‚) β†’ βˆƒ!β„Ž β„Ž ∈ (𝑂𝐻𝑂)))
98adantl 480 . . . 4 (((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (βˆ€π‘œ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘œπ»π‘‚) β†’ βˆƒ!β„Ž β„Ž ∈ (𝑂𝐻𝑂)))
10 eusn 4739 . . . . 5 (βˆƒ!β„Ž β„Ž ∈ (𝑂𝐻𝑂) ↔ βˆƒβ„Ž(𝑂𝐻𝑂) = {β„Ž})
11 eqid 2728 . . . . . . . . 9 (Idβ€˜πΆ) = (Idβ€˜πΆ)
123ad2antrr 724 . . . . . . . . 9 (((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ 𝐢 ∈ Cat)
13 simpr 483 . . . . . . . . 9 (((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ 𝑂 ∈ 𝐡)
141, 2, 11, 12, 13catidcl 17669 . . . . . . . 8 (((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ ((Idβ€˜πΆ)β€˜π‘‚) ∈ (𝑂𝐻𝑂))
15 fvex 6915 . . . . . . . . . . . . 13 ((Idβ€˜πΆ)β€˜π‘‚) ∈ V
1615elsn 4647 . . . . . . . . . . . 12 (((Idβ€˜πΆ)β€˜π‘‚) ∈ {β„Ž} ↔ ((Idβ€˜πΆ)β€˜π‘‚) = β„Ž)
17 eqcom 2735 . . . . . . . . . . . 12 (((Idβ€˜πΆ)β€˜π‘‚) = β„Ž ↔ β„Ž = ((Idβ€˜πΆ)β€˜π‘‚))
18 sneqbg 4849 . . . . . . . . . . . . . 14 (β„Ž ∈ V β†’ ({β„Ž} = {((Idβ€˜πΆ)β€˜π‘‚)} ↔ β„Ž = ((Idβ€˜πΆ)β€˜π‘‚)))
1918bicomd 222 . . . . . . . . . . . . 13 (β„Ž ∈ V β†’ (β„Ž = ((Idβ€˜πΆ)β€˜π‘‚) ↔ {β„Ž} = {((Idβ€˜πΆ)β€˜π‘‚)}))
2019elv 3479 . . . . . . . . . . . 12 (β„Ž = ((Idβ€˜πΆ)β€˜π‘‚) ↔ {β„Ž} = {((Idβ€˜πΆ)β€˜π‘‚)})
2116, 17, 203bitri 296 . . . . . . . . . . 11 (((Idβ€˜πΆ)β€˜π‘‚) ∈ {β„Ž} ↔ {β„Ž} = {((Idβ€˜πΆ)β€˜π‘‚)})
2221biimpi 215 . . . . . . . . . 10 (((Idβ€˜πΆ)β€˜π‘‚) ∈ {β„Ž} β†’ {β„Ž} = {((Idβ€˜πΆ)β€˜π‘‚)})
2322a1i 11 . . . . . . . . 9 ((𝑂𝐻𝑂) = {β„Ž} β†’ (((Idβ€˜πΆ)β€˜π‘‚) ∈ {β„Ž} β†’ {β„Ž} = {((Idβ€˜πΆ)β€˜π‘‚)}))
24 eleq2 2818 . . . . . . . . 9 ((𝑂𝐻𝑂) = {β„Ž} β†’ (((Idβ€˜πΆ)β€˜π‘‚) ∈ (𝑂𝐻𝑂) ↔ ((Idβ€˜πΆ)β€˜π‘‚) ∈ {β„Ž}))
25 eqeq1 2732 . . . . . . . . 9 ((𝑂𝐻𝑂) = {β„Ž} β†’ ((𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)} ↔ {β„Ž} = {((Idβ€˜πΆ)β€˜π‘‚)}))
2623, 24, 253imtr4d 293 . . . . . . . 8 ((𝑂𝐻𝑂) = {β„Ž} β†’ (((Idβ€˜πΆ)β€˜π‘‚) ∈ (𝑂𝐻𝑂) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
2714, 26syl5 34 . . . . . . 7 ((𝑂𝐻𝑂) = {β„Ž} β†’ (((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
2827exlimiv 1925 . . . . . 6 (βˆƒβ„Ž(𝑂𝐻𝑂) = {β„Ž} β†’ (((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
2928com12 32 . . . . 5 (((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (βˆƒβ„Ž(𝑂𝐻𝑂) = {β„Ž} β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
3010, 29biimtrid 241 . . . 4 (((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (βˆƒ!β„Ž β„Ž ∈ (𝑂𝐻𝑂) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
319, 30syld 47 . . 3 (((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (βˆ€π‘œ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘œπ»π‘‚) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
3231expimpd 452 . 2 ((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) β†’ ((𝑂 ∈ 𝐡 ∧ βˆ€π‘œ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘œπ»π‘‚)) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
334, 32mpd 15 1 ((πœ‘ ∧ 𝑂 ∈ (TermOβ€˜πΆ)) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆƒ!weu 2557  βˆ€wral 3058  Vcvv 3473  {csn 4632  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  Hom chom 17251  Catccat 17651  Idccid 17652  TermOctermo 17978
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-cat 17655  df-cid 17656  df-termo 17981
This theorem is referenced by:  2termoinv  18013
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