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Theorem idmon 45824
Description: An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.)
Hypotheses
Ref Expression
idmon.b 𝐵 = (Base‘𝐶)
idmon.h 𝐻 = (Hom ‘𝐶)
idmon.i 1 = (Id‘𝐶)
idmon.c (𝜑𝐶 ∈ Cat)
idmon.x (𝜑𝑋𝐵)
idmon.m 𝑀 = (Mono‘𝐶)
Assertion
Ref Expression
idmon (𝜑 → ( 1𝑋) ∈ (𝑋𝑀𝑋))

Proof of Theorem idmon
Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idmon.b . . 3 𝐵 = (Base‘𝐶)
2 idmon.h . . 3 𝐻 = (Hom ‘𝐶)
3 idmon.i . . 3 1 = (Id‘𝐶)
4 idmon.c . . 3 (𝜑𝐶 ∈ Cat)
5 idmon.x . . 3 (𝜑𝑋𝐵)
61, 2, 3, 4, 5catidcl 17068 . 2 (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))
74adantr 484 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
8 simpr1 1195 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝑧𝐵)
9 eqid 2739 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
105adantr 484 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝑋𝐵)
11 simpr2 1196 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
121, 2, 3, 7, 8, 9, 10, 11catlid 17069 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = 𝑔)
13 simpr3 1197 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → ∈ (𝑧𝐻𝑋))
141, 2, 3, 7, 8, 9, 10, 13catlid 17069 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)) = )
1512, 14eqeq12d 2755 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → ((( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)) ↔ 𝑔 = ))
1615biimpd 232 . . 3 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → ((( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)) → 𝑔 = ))
1716ralrimivvva 3105 . 2 (𝜑 → ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)) → 𝑔 = ))
18 idmon.m . . 3 𝑀 = (Mono‘𝐶)
191, 2, 9, 18, 4, 5, 5ismon2 17121 . 2 (𝜑 → (( 1𝑋) ∈ (𝑋𝑀𝑋) ↔ (( 1𝑋) ∈ (𝑋𝐻𝑋) ∧ ∀𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋)∀ ∈ (𝑧𝐻𝑋)((( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)) → 𝑔 = ))))
206, 17, 19mpbir2and 713 1 (𝜑 → ( 1𝑋) ∈ (𝑋𝑀𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  cop 4532  cfv 6349  (class class class)co 7182  Basecbs 16598  Hom chom 16691  compcco 16692  Catccat 17050  Idccid 17051  Monocmon 17115
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-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
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-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  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-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-1st 7726  df-2nd 7727  df-cat 17054  df-cid 17055  df-mon 17117
This theorem is referenced by: (None)
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