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Theorem fthmon 17887
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 2726 . . . . . 6 (Baseβ€˜π·) = (Baseβ€˜π·)
3 eqid 2726 . . . . . 6 (Hom β€˜π·) = (Hom β€˜π·)
4 eqid 2726 . . . . . 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 17867 . . . . . . . . . . . 12 (𝐢 Faith 𝐷) βŠ† (𝐢 Func 𝐷)
87ssbri 5186 . . . . . . . . . . 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 5142 . . . . . . . . . 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 17820 . . . . . . . . 9 (⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷) β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
1311, 12syl 17 . . . . . . . 8 (πœ‘ β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
1413simprd 495 . . . . . . 7 (πœ‘ β†’ 𝐷 ∈ Cat)
1514adantr 480 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝐷 ∈ Cat)
16 fthmon.b . . . . . . . 8 𝐡 = (Baseβ€˜πΆ)
179adantr 480 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝐹(𝐢 Func 𝐷)𝐺)
1816, 2, 17funcf1 17823 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝐹:𝐡⟢(Baseβ€˜π·))
19 fthmon.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ 𝐡)
2019adantr 480 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝑋 ∈ 𝐡)
2118, 20ffvelcdmd 7080 . . . . . 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 480 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ π‘Œ ∈ 𝐡)
2418, 23ffvelcdmd 7080 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (πΉβ€˜π‘Œ) ∈ (Baseβ€˜π·))
25 simpr1 1191 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝑧 ∈ 𝐡)
2618, 25ffvelcdmd 7080 . . . . . 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 480 . . . . . 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 17825 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (𝑧𝐺𝑋):(𝑧𝐻𝑋)⟢((πΉβ€˜π‘§)(Hom β€˜π·)(πΉβ€˜π‘‹)))
31 simpr2 1192 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝑓 ∈ (𝑧𝐻𝑋))
3230, 31ffvelcdmd 7080 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ ((𝑧𝐺𝑋)β€˜π‘“) ∈ ((πΉβ€˜π‘§)(Hom β€˜π·)(πΉβ€˜π‘‹)))
33 simpr3 1193 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝑔 ∈ (𝑧𝐻𝑋))
3430, 33ffvelcdmd 7080 . . . . . 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 17690 . . . . 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 2726 . . . . . . . 8 (compβ€˜πΆ) = (compβ€˜πΆ)
371adantr 480 . . . . . . . 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 17828 . . . . . . 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 17828 . . . . . . 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 2742 . . . . . 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 480 . . . . . . 7 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝐹(𝐢 Faith 𝐷)𝐺)
4213simpld 494 . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ Cat)
4342adantr 480 . . . . . . . 8 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ 𝐢 ∈ Cat)
4416, 29, 36, 43, 25, 20, 23, 31, 37catcocl 17636 . . . . . . 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 17636 . . . . . . 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 17878 . . . . . 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 281 . . . . 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 17878 . . . . 5 ((πœ‘ ∧ (𝑧 ∈ 𝐡 ∧ 𝑓 ∈ (𝑧𝐻𝑋) ∧ 𝑔 ∈ (𝑧𝐻𝑋))) β†’ (((𝑧𝐺𝑋)β€˜π‘“) = ((𝑧𝐺𝑋)β€˜π‘”) ↔ 𝑓 = 𝑔))
4935, 47, 483bitr3d 309 . . . 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 3197 . 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 17688 . 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 710 1 (πœ‘ β†’ 𝑅 ∈ (π‘‹π‘€π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βŸ¨cop 4629   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  Hom chom 17215  compcco 17216  Catccat 17615  Monocmon 17682   Func cfunc 17811   Faith cfth 17863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-ixp 8891  df-cat 17619  df-mon 17684  df-func 17815  df-fth 17865
This theorem is referenced by:  fthepi  17888
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