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Theorem fthmon 17874
Description: A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fthmon.b 𝐡 = (Baseβ€˜πΆ)
fthmon.h 𝐻 = (Hom β€˜πΆ)
fthmon.f (πœ‘ β†’ 𝐹(𝐢 Faith 𝐷)𝐺)
fthmon.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
fthmon.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
fthmon.r (πœ‘ β†’ 𝑅 ∈ (π‘‹π»π‘Œ))
fthmon.m 𝑀 = (Monoβ€˜πΆ)
fthmon.n 𝑁 = (Monoβ€˜π·)
fthmon.1 (πœ‘ β†’ ((π‘‹πΊπ‘Œ)β€˜π‘…) ∈ ((πΉβ€˜π‘‹)𝑁(πΉβ€˜π‘Œ)))
Assertion
Ref Expression
fthmon (πœ‘ β†’ 𝑅 ∈ (π‘‹π‘€π‘Œ))
Proof of Theorem fthmon
Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fthmon.r . 2 (πœ‘ β†’ 𝑅 ∈ (π‘‹π»π‘Œ))
2 eqid 2732 . . . . . 6 (Baseβ€˜π·) = (Baseβ€˜π·)
3 eqid 2732 . . . . . 6 (Hom β€˜π·) = (Hom β€˜π·)
4 eqid 2732 . . . . . 6 (compβ€˜π·) = (compβ€˜π·)
5 fthmon.n . . . . . 6 𝑁 = (Monoβ€˜π·)
6 fthmon.f . . . . . . . . . . 11 (πœ‘ β†’ 𝐹(𝐢 Faith 𝐷)𝐺)
7 fthfunc 17854 . . . . . . . . . . . 12 (𝐢 Faith 𝐷) βŠ† (𝐢 Func 𝐷)
87ssbri 5192 . . . . . . . . . . 11 (𝐹(𝐢 Faith 𝐷)𝐺 β†’ 𝐹(𝐢 Func 𝐷)𝐺)
96, 8syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐹(𝐢 Func 𝐷)𝐺)
10 df-br 5148 . . . . . . . . . 10 (𝐹(𝐢 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷))
119, 10sylib 217 . . . . . . . . 9 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷))
12 funcrcl 17809 . . . . . . . . 9 (⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷) β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . . . 8 (πœ‘ β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simprd 496 . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ Cat)
1514adantr 481 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝐷 ∈ Cat)
16 fthmon.b . . . . . . . 8 𝐡 = (Baseβ€˜πΆ)
179adantr 481 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝐹(𝐢 Func 𝐷)𝐺)
1816, 2, 17funcf1 17812 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝐹:𝐡⟢(Baseβ€˜π·))
19 fthmon.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ 𝐡)
2019adantr 481 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝑋 ∈ 𝐡)
2118, 20ffvelcdmd 7084 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (πΉβ€˜π‘‹) ∈ (Baseβ€˜π·))
22 fthmon.y . . . . . . . 8 (πœ‘ β†’ π‘Œ ∈ 𝐡)
2322adantr 481 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ π‘Œ ∈ 𝐡)
2418, 23ffvelcdmd 7084 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (πΉβ€˜π‘Œ) ∈ (Baseβ€˜π·))
25 simpr1 1194 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝑧 ∈ 𝐡)
2618, 25ffvelcdmd 7084 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (πΉβ€˜π‘§) ∈ (Baseβ€˜π·))
27 fthmon.1 . . . . . . 7 (πœ‘ β†’ ((π‘‹πΊπ‘Œ)β€˜π‘…) ∈ ((πΉβ€˜π‘‹)𝑁(πΉβ€˜π‘Œ)))
2827adantr 481 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ ((π‘‹πΊπ‘Œ)β€˜π‘…) ∈ ((πΉβ€˜π‘‹)𝑁(πΉβ€˜π‘Œ)))
29 fthmon.h . . . . . . . 8 𝐻 = (Hom β€˜πΆ)
3016, 29, 3, 17, 25, 20funcf2 17814 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (𝑧𝐺𝑋):(𝑧𝐻𝑋)⟢((πΉβ€˜π‘§)(Hom β€˜π·)(πΉβ€˜π‘‹)))
31 simpr2 1195 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝑓 ∈ (𝑧𝐻𝑋))
3230, 31ffvelcdmd 7084 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ ((𝑧𝐺𝑋)β€˜π‘“) ∈ ((πΉβ€˜π‘§)(Hom β€˜π·)(πΉβ€˜π‘‹)))
33 simpr3 1196 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝑔 ∈ (𝑧𝐻𝑋))
3430, 33ffvelcdmd 7084 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ ((𝑧𝐺𝑋)β€˜π‘”) ∈ ((πΉβ€˜π‘§)(Hom β€˜π·)(πΉβ€˜π‘‹)))
352, 3, 4, 5, 15, 21, 24, 26, 28, 32, 34moni 17679 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ ((((π‘‹πΊπ‘Œ)β€˜π‘…)(⟨(πΉβ€˜π‘§), (πΉβ€˜π‘‹)⟩(compβ€˜π·)(πΉβ€˜π‘Œ))((𝑧𝐺𝑋)β€˜π‘“)) = (((π‘‹πΊπ‘Œ)β€˜π‘…)(⟨(πΉβ€˜π‘§), (πΉβ€˜π‘‹)⟩(compβ€˜π·)(πΉβ€˜π‘Œ))((𝑧𝐺𝑋)β€˜π‘”)) ↔ ((𝑧𝐺𝑋)β€˜π‘“) = ((𝑧𝐺𝑋)β€˜π‘”)))
36 eqid 2732 . . . . . . . 8 (compβ€˜πΆ) = (compβ€˜πΆ)
371adantr 481 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝑅 ∈ (π‘‹π»π‘Œ))
3816, 29, 36, 4, 17, 25, 20, 23, 31, 37funcco 17817 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ ((π‘§πΊπ‘Œ)β€˜(𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑓)) = (((π‘‹πΊπ‘Œ)β€˜π‘…)(⟨(πΉβ€˜π‘§), (πΉβ€˜π‘‹)⟩(compβ€˜π·)(πΉβ€˜π‘Œ))((𝑧𝐺𝑋)β€˜π‘“)))
3916, 29, 36, 4, 17, 25, 20, 23, 33, 37funcco 17817 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ ((π‘§πΊπ‘Œ)β€˜(𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔)) = (((π‘‹πΊπ‘Œ)β€˜π‘…)(⟨(πΉβ€˜π‘§), (πΉβ€˜π‘‹)⟩(compβ€˜π·)(πΉβ€˜π‘Œ))((𝑧𝐺𝑋)β€˜π‘”)))
4038, 39eqeq12d 2748 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (((π‘§πΊπ‘Œ)β€˜(𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑓)) = ((π‘§πΊπ‘Œ)β€˜(𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔)) ↔ (((π‘‹πΊπ‘Œ)β€˜π‘…)(⟨(πΉβ€˜π‘§), (πΉβ€˜π‘‹)⟩(compβ€˜π·)(πΉβ€˜π‘Œ))((𝑧𝐺𝑋)β€˜π‘“)) = (((π‘‹πΊπ‘Œ)β€˜π‘…)(⟨(πΉβ€˜π‘§), (πΉβ€˜π‘‹)⟩(compβ€˜π·)(πΉβ€˜π‘Œ))((𝑧𝐺𝑋)β€˜π‘”))))
416adantr 481 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝐹(𝐢 Faith 𝐷)𝐺)
4213simpld 495 . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ Cat)
4342adantr 481 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝐢 ∈ Cat)
4416, 29, 36, 43, 25, 20, 23, 31, 37catcocl 17625 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑓) ∈ (π‘§π»π‘Œ))
4516, 29, 36, 43, 25, 20, 23, 33, 37catcocl 17625 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) ∈ (π‘§π»π‘Œ))
4616, 29, 3, 41, 25, 23, 44, 45fthi 17865 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (((π‘§πΊπ‘Œ)β€˜(𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑓)) = ((π‘§πΊπ‘Œ)β€˜(𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔)) ↔ (𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑓) = (𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔)))
4740, 46bitr3d 280 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ ((((π‘‹πΊπ‘Œ)β€˜π‘…)(⟨(πΉβ€˜π‘§), (πΉβ€˜π‘‹)⟩(compβ€˜π·)(πΉβ€˜π‘Œ))((𝑧𝐺𝑋)β€˜π‘“)) = (((π‘‹πΊπ‘Œ)β€˜π‘…)(⟨(πΉβ€˜π‘§), (πΉβ€˜π‘‹)⟩(compβ€˜π·)(πΉβ€˜π‘Œ))((𝑧𝐺𝑋)β€˜π‘”)) ↔ (𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑓) = (𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔)))
4816, 29, 3, 41, 25, 20, 31, 33fthi 17865 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (((𝑧𝐺𝑋)β€˜π‘“) = ((𝑧𝐺𝑋)β€˜π‘”) ↔ 𝑓 = 𝑔))
4935, 47, 483bitr3d 308 . . . 4 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ ((𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑓) = (𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) ↔ 𝑓 = 𝑔))
5049biimpd 228 . . 3 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ ((𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑓) = (𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) β†’ 𝑓 = 𝑔))
5150ralrimivvva 3203 . 2 (πœ‘ β†’ βˆ€π‘§ ∈ 𝐡 βˆ€π‘“ ∈ (𝑧𝐻𝑋)βˆ€π‘” ∈ (𝑧𝐻𝑋)((𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑓) = (𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) β†’ 𝑓 = 𝑔))
52 fthmon.m . . 3 𝑀 = (Monoβ€˜πΆ)
5316, 29, 36, 52, 42, 19, 22ismon2 17677 . 2 (πœ‘ β†’ (𝑅 ∈ (π‘‹π‘€π‘Œ) ↔ (𝑅 ∈ (π‘‹π»π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 βˆ€π‘“ ∈ (𝑧𝐻𝑋)βˆ€π‘” ∈ (𝑧𝐻𝑋)((𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑓) = (𝑅(βŸ¨π‘§, π‘‹βŸ©(compβ€˜πΆ)π‘Œ)𝑔) β†’ 𝑓 = 𝑔))))
541, 51, 53mpbir2and 711 1 (πœ‘ β†’ 𝑅 ∈ (π‘‹π‘€π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βŸ¨cop 4633   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  Hom chom 17204  compcco 17205  Catccat 17604  Monocmon 17671   Func cfunc 17800   Faith cfth 17850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-ixp 8888  df-cat 17608  df-mon 17673  df-func 17804  df-fth 17852
This theorem is referenced by:  fthepi  17875
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