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Theorem thincmo 45833
Description: There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.)
Hypotheses
Ref Expression
thincmo.c (𝜑𝐶 ∈ ThinCat)
thincmo.x (𝜑𝑋𝐵)
thincmo.y (𝜑𝑌𝐵)
thincmo.b 𝐵 = (Base‘𝐶)
thincmo.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
thincmo (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Distinct variable groups:   𝐵,𝑓   𝐶,𝑓   𝑓,𝐻   𝑓,𝑋   𝑓,𝑌   𝜑,𝑓

Proof of Theorem thincmo
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 thincmo.x . . . . . 6 (𝜑𝑋𝐵)
21adantr 484 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑋𝐵)
3 thincmo.y . . . . . 6 (𝜑𝑌𝐵)
43adantr 484 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑌𝐵)
5 simprl 771 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 ∈ (𝑋𝐻𝑌))
6 simprr 773 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑔 ∈ (𝑋𝐻𝑌))
7 thincmo.b . . . . 5 𝐵 = (Base‘𝐶)
8 thincmo.h . . . . 5 𝐻 = (Hom ‘𝐶)
9 thincmo.c . . . . . 6 (𝜑𝐶 ∈ ThinCat)
109adantr 484 . . . . 5 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝐶 ∈ ThinCat)
112, 4, 5, 6, 7, 8, 10thincmo2 45832 . . . 4 ((𝜑 ∧ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌))) → 𝑓 = 𝑔)
1211ex 416 . . 3 (𝜑 → ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
1312alrimivv 1935 . 2 (𝜑 → ∀𝑓𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
14 eleq1w 2816 . . 3 (𝑓 = 𝑔 → (𝑓 ∈ (𝑋𝐻𝑌) ↔ 𝑔 ∈ (𝑋𝐻𝑌)))
1514mo4 2567 . 2 (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌) ↔ ∀𝑓𝑔((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝑔))
1613, 15sylibr 237 1 (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1540   = wceq 1542  wcel 2114  ∃*wmo 2539  cfv 6349  (class class class)co 7182  Basecbs 16598  Hom chom 16691  ThinCatcthinc 45826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-nul 5184
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6307  df-fv 6357  df-ov 7185  df-thinc 45827
This theorem is referenced by:  thincmod  45835
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