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Theorem fthmon 16808
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 2760 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
3 eqid 2760 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
4 eqid 2760 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
5 fthmon.n . . . . . 6 𝑁 = (Mono‘𝐷)
6 fthmon.f . . . . . . . . . . 11 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
7 fthfunc 16788 . . . . . . . . . . . 12 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
87ssbri 4849 . . . . . . . . . . 11 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
96, 8syl 17 . . . . . . . . . 10 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
10 df-br 4805 . . . . . . . . . 10 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylib 208 . . . . . . . . 9 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12 funcrcl 16744 . . . . . . . . 9 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . . . 8 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simprd 482 . . . . . . 7 (𝜑𝐷 ∈ Cat)
1514adantr 472 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐷 ∈ Cat)
16 fthmon.b . . . . . . . 8 𝐵 = (Base‘𝐶)
179adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Func 𝐷)𝐺)
1816, 2, 17funcf1 16747 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹:𝐵⟶(Base‘𝐷))
19 fthmon.x . . . . . . . 8 (𝜑𝑋𝐵)
2019adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑋𝐵)
2118, 20ffvelrnd 6524 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑋) ∈ (Base‘𝐷))
22 fthmon.y . . . . . . . 8 (𝜑𝑌𝐵)
2322adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑌𝐵)
2418, 23ffvelrnd 6524 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑌) ∈ (Base‘𝐷))
25 simpr1 1234 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑧𝐵)
2618, 25ffvelrnd 6524 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑧) ∈ (Base‘𝐷))
27 fthmon.1 . . . . . . 7 (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))
2827adantr 472 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))
29 fthmon.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
3016, 29, 3, 17, 25, 20funcf2 16749 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑧𝐺𝑋):(𝑧𝐻𝑋)⟶((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
31 simpr2 1236 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑓 ∈ (𝑧𝐻𝑋))
3230, 31ffvelrnd 6524 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑓) ∈ ((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
33 simpr3 1238 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
3430, 33ffvelrnd 6524 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑔) ∈ ((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
352, 3, 4, 5, 15, 21, 24, 26, 28, 32, 34moni 16617 . . . . 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 2760 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
371adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑅 ∈ (𝑋𝐻𝑌))
3816, 29, 36, 4, 17, 25, 20, 23, 31, 37funcco 16752 . . . . . . 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 16752 . . . . . . 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 2775 . . . . . 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 472 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Faith 𝐷)𝐺)
4213simpld 477 . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
4342adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
4416, 29, 36, 43, 25, 20, 23, 31, 37catcocl 16567 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) ∈ (𝑧𝐻𝑌))
4516, 29, 36, 43, 25, 20, 23, 33, 37catcocl 16567 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) ∈ (𝑧𝐻𝑌))
4616, 29, 3, 41, 25, 23, 44, 45fthi 16799 . . . . . 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 270 . . . . 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 16799 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑋)‘𝑓) = ((𝑧𝐺𝑋)‘𝑔) ↔ 𝑓 = 𝑔))
4935, 47, 483bitr3d 298 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) ↔ 𝑓 = 𝑔))
5049biimpd 219 . . 3 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))
5150ralrimivvva 3110 . 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 16615 . 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 995 1 (𝜑𝑅 ∈ (𝑋𝑀𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  cop 4327   class class class wbr 4804  cfv 6049  (class class class)co 6814  Basecbs 16079  Hom chom 16174  compcco 16175  Catccat 16546  Monocmon 16609   Func cfunc 16735   Faith cfth 16784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-ixp 8077  df-cat 16550  df-mon 16611  df-func 16739  df-fth 16786
This theorem is referenced by:  fthepi  16809
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