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Theorem initoid 17995
Description: For an initial object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 6-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
initoid ((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)})

Proof of Theorem initoid
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, 3isinitoi 17993 . 2 ((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) β†’ (𝑂 ∈ 𝐡 ∧ βˆ€π‘œ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘‚π»π‘œ)))
5 oveq2 7432 . . . . . . . 8 (π‘œ = 𝑂 β†’ (π‘‚π»π‘œ) = (𝑂𝐻𝑂))
65eleq2d 2814 . . . . . . 7 (π‘œ = 𝑂 β†’ (β„Ž ∈ (π‘‚π»π‘œ) ↔ β„Ž ∈ (𝑂𝐻𝑂)))
76eubidv 2575 . . . . . 6 (π‘œ = 𝑂 β†’ (βˆƒ!β„Ž β„Ž ∈ (π‘‚π»π‘œ) ↔ βˆƒ!β„Ž β„Ž ∈ (𝑂𝐻𝑂)))
87rspcv 3605 . . . . 5 (𝑂 ∈ 𝐡 β†’ (βˆ€π‘œ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘‚π»π‘œ) β†’ βˆƒ!β„Ž β„Ž ∈ (𝑂𝐻𝑂)))
98adantl 480 . . . 4 (((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (βˆ€π‘œ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘‚π»π‘œ) β†’ βˆƒ!β„Ž β„Ž ∈ (𝑂𝐻𝑂)))
10 eusn 4737 . . . . 5 (βˆƒ!β„Ž β„Ž ∈ (𝑂𝐻𝑂) ↔ βˆƒβ„Ž(𝑂𝐻𝑂) = {β„Ž})
11 eqid 2727 . . . . . . . . 9 (Idβ€˜πΆ) = (Idβ€˜πΆ)
123ad2antrr 724 . . . . . . . . 9 (((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ 𝐢 ∈ Cat)
13 simpr 483 . . . . . . . . 9 (((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ 𝑂 ∈ 𝐡)
141, 2, 11, 12, 13catidcl 17667 . . . . . . . 8 (((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ ((Idβ€˜πΆ)β€˜π‘‚) ∈ (𝑂𝐻𝑂))
15 fvex 6913 . . . . . . . . . . . . 13 ((Idβ€˜πΆ)β€˜π‘‚) ∈ V
1615elsn 4645 . . . . . . . . . . . 12 (((Idβ€˜πΆ)β€˜π‘‚) ∈ {β„Ž} ↔ ((Idβ€˜πΆ)β€˜π‘‚) = β„Ž)
17 eqcom 2734 . . . . . . . . . . . 12 (((Idβ€˜πΆ)β€˜π‘‚) = β„Ž ↔ β„Ž = ((Idβ€˜πΆ)β€˜π‘‚))
18 sneqbg 4847 . . . . . . . . . . . . . 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 3477 . . . . . . . . . . . 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 2817 . . . . . . . . 9 ((𝑂𝐻𝑂) = {β„Ž} β†’ (((Idβ€˜πΆ)β€˜π‘‚) ∈ (𝑂𝐻𝑂) ↔ ((Idβ€˜πΆ)β€˜π‘‚) ∈ {β„Ž}))
25 eqeq1 2731 . . . . . . . . 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 ∈ (InitOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
2827exlimiv 1925 . . . . . 6 (βˆƒβ„Ž(𝑂𝐻𝑂) = {β„Ž} β†’ (((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
2928com12 32 . . . . 5 (((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (βˆƒβ„Ž(𝑂𝐻𝑂) = {β„Ž} β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
3010, 29biimtrid 241 . . . 4 (((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜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 ∈ (InitOβ€˜πΆ)) ∧ 𝑂 ∈ 𝐡) β†’ (βˆ€π‘œ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘‚π»π‘œ) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
3231expimpd 452 . 2 ((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜πΆ)) β†’ ((𝑂 ∈ 𝐡 ∧ βˆ€π‘œ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘‚π»π‘œ)) β†’ (𝑂𝐻𝑂) = {((Idβ€˜πΆ)β€˜π‘‚)}))
334, 32mpd 15 1 ((πœ‘ ∧ 𝑂 ∈ (InitOβ€˜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 3057  Vcvv 3471  {csn 4630  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  Hom chom 17249  Catccat 17649  Idccid 17650  InitOcinito 17975
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-cat 17653  df-cid 17654  df-inito 17978
This theorem is referenced by:  2initoinv  18004
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