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Theorem fthmon 16410
 Description: A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fthmon.b 𝐵 = (Base‘𝐶)
fthmon.h 𝐻 = (Hom ‘𝐶)
fthmon.f (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
fthmon.x (𝜑𝑋𝐵)
fthmon.y (𝜑𝑌𝐵)
fthmon.r (𝜑𝑅 ∈ (𝑋𝐻𝑌))
fthmon.m 𝑀 = (Mono‘𝐶)
fthmon.n 𝑁 = (Mono‘𝐷)
fthmon.1 (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))
Assertion
Ref Expression
fthmon (𝜑𝑅 ∈ (𝑋𝑀𝑌))

Proof of Theorem fthmon
Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fthmon.r . 2 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
2 eqid 2610 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
3 eqid 2610 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
4 eqid 2610 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
5 fthmon.n . . . . . 6 𝑁 = (Mono‘𝐷)
6 fthmon.f . . . . . . . . . . 11 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
7 fthfunc 16390 . . . . . . . . . . . 12 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
87ssbri 4627 . . . . . . . . . . 11 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
96, 8syl 17 . . . . . . . . . 10 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
10 df-br 4584 . . . . . . . . . 10 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylib 207 . . . . . . . . 9 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12 funcrcl 16346 . . . . . . . . 9 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . . . 8 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simprd 478 . . . . . . 7 (𝜑𝐷 ∈ Cat)
1514adantr 480 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐷 ∈ Cat)
16 fthmon.b . . . . . . . 8 𝐵 = (Base‘𝐶)
179adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Func 𝐷)𝐺)
1816, 2, 17funcf1 16349 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹:𝐵⟶(Base‘𝐷))
19 fthmon.x . . . . . . . 8 (𝜑𝑋𝐵)
2019adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑋𝐵)
2118, 20ffvelrnd 6268 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑋) ∈ (Base‘𝐷))
22 fthmon.y . . . . . . . 8 (𝜑𝑌𝐵)
2322adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑌𝐵)
2418, 23ffvelrnd 6268 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑌) ∈ (Base‘𝐷))
25 simpr1 1060 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑧𝐵)
2618, 25ffvelrnd 6268 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑧) ∈ (Base‘𝐷))
27 fthmon.1 . . . . . . 7 (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))
2827adantr 480 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))
29 fthmon.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
3016, 29, 3, 17, 25, 20funcf2 16351 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑧𝐺𝑋):(𝑧𝐻𝑋)⟶((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
31 simpr2 1061 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑓 ∈ (𝑧𝐻𝑋))
3230, 31ffvelrnd 6268 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑓) ∈ ((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
33 simpr3 1062 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
3430, 33ffvelrnd 6268 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑔) ∈ ((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
352, 3, 4, 5, 15, 21, 24, 26, 28, 32, 34moni 16219 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑔)) ↔ ((𝑧𝐺𝑋)‘𝑓) = ((𝑧𝐺𝑋)‘𝑔)))
36 eqid 2610 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
371adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑅 ∈ (𝑋𝐻𝑌))
3816, 29, 36, 4, 17, 25, 20, 23, 31, 37funcco 16354 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑓)))
3916, 29, 36, 4, 17, 25, 20, 23, 33, 37funcco 16354 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑔)))
4038, 39eqeq12d 2625 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓)) = ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) ↔ (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑔))))
416adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Faith 𝐷)𝐺)
4213simpld 474 . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
4342adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
4416, 29, 36, 43, 25, 20, 23, 31, 37catcocl 16169 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) ∈ (𝑧𝐻𝑌))
4516, 29, 36, 43, 25, 20, 23, 33, 37catcocl 16169 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) ∈ (𝑧𝐻𝑌))
4616, 29, 3, 41, 25, 23, 44, 45fthi 16401 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓)) = ((𝑧𝐺𝑌)‘(𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)) ↔ (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)))
4740, 46bitr3d 269 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑓)) = (((𝑋𝐺𝑌)‘𝑅)(⟨(𝐹𝑧), (𝐹𝑋)⟩(comp‘𝐷)(𝐹𝑌))((𝑧𝐺𝑋)‘𝑔)) ↔ (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔)))
4816, 29, 3, 41, 25, 20, 31, 33fthi 16401 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑋)‘𝑓) = ((𝑧𝐺𝑋)‘𝑔) ↔ 𝑓 = 𝑔))
4935, 47, 483bitr3d 297 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) ↔ 𝑓 = 𝑔))
5049biimpd 218 . . 3 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))
5150ralrimivvva 2955 . 2 (𝜑 → ∀𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋)∀𝑔 ∈ (𝑧𝐻𝑋)((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))
52 fthmon.m . . 3 𝑀 = (Mono‘𝐶)
5316, 29, 36, 52, 42, 19, 22ismon2 16217 . 2 (𝜑 → (𝑅 ∈ (𝑋𝑀𝑌) ↔ (𝑅 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋)∀𝑔 ∈ (𝑧𝐻𝑋)((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))))
541, 51, 53mpbir2and 959 1 (𝜑𝑅 ∈ (𝑋𝑀𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Monocmon 16211   Func cfunc 16337   Faith cfth 16386 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-cat 16152  df-mon 16213  df-func 16341  df-fth 16388 This theorem is referenced by:  fthepi  16411
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