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Theorem idmon 47538
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 17622 . 2 (𝜑 → ( 1𝑋) ∈ (𝑋𝐻𝑋))
74adantr 482 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
8 simpr1 1195 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝑧𝐵)
9 eqid 2733 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
105adantr 482 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝑋𝐵)
11 simpr2 1196 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
121, 2, 3, 7, 8, 9, 10, 11catlid 17623 . . . . 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 17623 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑔 ∈ (𝑧𝐻𝑋) ∧ ∈ (𝑧𝐻𝑋))) → (( 1𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)) = )
1512, 14eqeq12d 2749 . . . 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 3204 . 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 17677 . 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 712 1 (𝜑 → ( 1𝑋) ∈ (𝑋𝑀𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  cop 4633  cfv 6540  (class class class)co 7404  Basecbs 17140  Hom chom 17204  compcco 17205  Catccat 17604  Idccid 17605  Monocmon 17671
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-cat 17608  df-cid 17609  df-mon 17673
This theorem is referenced by: (None)
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