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Theorem idmon 46297
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 17391 . 2 (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))
74adantr 481 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
8 simpr1 1193 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝑧𝐵)
9 eqid 2738 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
105adantr 481 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝑋𝐵)
11 simpr2 1194 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
121, 2, 3, 7, 8, 9, 10, 11catlid 17392 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = 𝑔)
13 simpr3 1195 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → ∈ (𝑧𝐻𝑋))
141, 2, 3, 7, 8, 9, 10, 13catlid 17392 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)) = )
1512, 14eqeq12d 2754 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → ((( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)) ↔ 𝑔 = ))
1615biimpd 228 . . 3 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → ((( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)) → 𝑔 = ))
1716ralrimivvva 3127 . 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 17446 . 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 710 1 (𝜑 → ( 1𝑋) ∈ (𝑋𝑀𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  cop 4567  cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Idccid 17374  Monocmon 17440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-cat 17377  df-cid 17378  df-mon 17442
This theorem is referenced by: (None)
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