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Theorem fthmon 16356
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 2609 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
3 eqid 2609 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
4 eqid 2609 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
5 fthmon.n . . . . . 6 𝑁 = (Mono‘𝐷)
6 fthmon.f . . . . . . . . . . 11 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
7 fthfunc 16336 . . . . . . . . . . . 12 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
87ssbri 4621 . . . . . . . . . . 11 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
96, 8syl 17 . . . . . . . . . 10 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
10 df-br 4578 . . . . . . . . . 10 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
119, 10sylib 206 . . . . . . . . 9 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
12 funcrcl 16292 . . . . . . . . 9 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . . . 8 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simprd 477 . . . . . . 7 (𝜑𝐷 ∈ Cat)
1514adantr 479 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐷 ∈ Cat)
16 fthmon.b . . . . . . . 8 𝐵 = (Base‘𝐶)
179adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Func 𝐷)𝐺)
1816, 2, 17funcf1 16295 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹:𝐵⟶(Base‘𝐷))
19 fthmon.x . . . . . . . 8 (𝜑𝑋𝐵)
2019adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑋𝐵)
2118, 20ffvelrnd 6253 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑋) ∈ (Base‘𝐷))
22 fthmon.y . . . . . . . 8 (𝜑𝑌𝐵)
2322adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑌𝐵)
2418, 23ffvelrnd 6253 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑌) ∈ (Base‘𝐷))
25 simpr1 1059 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑧𝐵)
2618, 25ffvelrnd 6253 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝐹𝑧) ∈ (Base‘𝐷))
27 fthmon.1 . . . . . . 7 (𝜑 → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))
2827adantr 479 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝑁(𝐹𝑌)))
29 fthmon.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
3016, 29, 3, 17, 25, 20funcf2 16297 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑧𝐺𝑋):(𝑧𝐻𝑋)⟶((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
31 simpr2 1060 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑓 ∈ (𝑧𝐻𝑋))
3230, 31ffvelrnd 6253 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑓) ∈ ((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
33 simpr3 1061 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
3430, 33ffvelrnd 6253 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑧𝐺𝑋)‘𝑔) ∈ ((𝐹𝑧)(Hom ‘𝐷)(𝐹𝑋)))
352, 3, 4, 5, 15, 21, 24, 26, 28, 32, 34moni 16165 . . . . 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 2609 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
371adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝑅 ∈ (𝑋𝐻𝑌))
3816, 29, 36, 4, 17, 25, 20, 23, 31, 37funcco 16300 . . . . . . 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 16300 . . . . . . 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 2624 . . . . . 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 479 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐹(𝐶 Faith 𝐷)𝐺)
4213simpld 473 . . . . . . . . 9 (𝜑𝐶 ∈ Cat)
4342adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
4416, 29, 36, 43, 25, 20, 23, 31, 37catcocl 16115 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) ∈ (𝑧𝐻𝑌))
4516, 29, 36, 43, 25, 20, 23, 33, 37catcocl 16115 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) ∈ (𝑧𝐻𝑌))
4616, 29, 3, 41, 25, 23, 44, 45fthi 16347 . . . . . 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 268 . . . . 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 16347 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → (((𝑧𝐺𝑋)‘𝑓) = ((𝑧𝐺𝑋)‘𝑔) ↔ 𝑓 = 𝑔))
4935, 47, 483bitr3d 296 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) ↔ 𝑓 = 𝑔))
5049biimpd 217 . . 3 ((𝜑 ∧ (𝑧𝐵𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) → ((𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑓) = (𝑅(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) → 𝑓 = 𝑔))
5150ralrimivvva 2954 . 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 16163 . 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 958 1 (𝜑𝑅 ∈ (𝑋𝑀𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  cop 4130   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  Hom chom 15725  compcco 15726  Catccat 16094  Monocmon 16157   Func cfunc 16283   Faith cfth 16332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723  df-ixp 7772  df-cat 16098  df-mon 16159  df-func 16287  df-fth 16334
This theorem is referenced by:  fthepi  16357
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