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Theorem moni 17684
Description: Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
ismon.b 𝐡 = (Baseβ€˜πΆ)
ismon.h 𝐻 = (Hom β€˜πΆ)
ismon.o Β· = (compβ€˜πΆ)
ismon.s 𝑀 = (Monoβ€˜πΆ)
ismon.c (πœ‘ β†’ 𝐢 ∈ Cat)
ismon.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
ismon.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
moni.z (πœ‘ β†’ 𝑍 ∈ 𝐡)
moni.f (πœ‘ β†’ 𝐹 ∈ (π‘‹π‘€π‘Œ))
moni.g (πœ‘ β†’ 𝐺 ∈ (𝑍𝐻𝑋))
moni.k (πœ‘ β†’ 𝐾 ∈ (𝑍𝐻𝑋))
Assertion
Ref Expression
moni (πœ‘ β†’ ((𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐺) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐾) ↔ 𝐺 = 𝐾))

Proof of Theorem moni
Dummy variables 𝑔 β„Ž 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 moni.f . . . . 5 (πœ‘ β†’ 𝐹 ∈ (π‘‹π‘€π‘Œ))
2 ismon.b . . . . . 6 𝐡 = (Baseβ€˜πΆ)
3 ismon.h . . . . . 6 𝐻 = (Hom β€˜πΆ)
4 ismon.o . . . . . 6 Β· = (compβ€˜πΆ)
5 ismon.s . . . . . 6 𝑀 = (Monoβ€˜πΆ)
6 ismon.c . . . . . 6 (πœ‘ β†’ 𝐢 ∈ Cat)
7 ismon.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ 𝐡)
8 ismon.y . . . . . 6 (πœ‘ β†’ π‘Œ ∈ 𝐡)
92, 3, 4, 5, 6, 7, 8ismon2 17682 . . . . 5 (πœ‘ β†’ (𝐹 ∈ (π‘‹π‘€π‘Œ) ↔ (𝐹 ∈ (π‘‹π»π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 βˆ€π‘” ∈ (𝑧𝐻𝑋)βˆ€β„Ž ∈ (𝑧𝐻𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))))
101, 9mpbid 231 . . . 4 (πœ‘ β†’ (𝐹 ∈ (π‘‹π»π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 βˆ€π‘” ∈ (𝑧𝐻𝑋)βˆ€β„Ž ∈ (𝑧𝐻𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)β„Ž) β†’ 𝑔 = β„Ž)))
1110simprd 495 . . 3 (πœ‘ β†’ βˆ€π‘§ ∈ 𝐡 βˆ€π‘” ∈ (𝑧𝐻𝑋)βˆ€β„Ž ∈ (𝑧𝐻𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
12 moni.z . . . 4 (πœ‘ β†’ 𝑍 ∈ 𝐡)
13 moni.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ (𝑍𝐻𝑋))
1413adantr 480 . . . . . 6 ((πœ‘ ∧ 𝑧 = 𝑍) β†’ 𝐺 ∈ (𝑍𝐻𝑋))
15 simpr 484 . . . . . . 7 ((πœ‘ ∧ 𝑧 = 𝑍) β†’ 𝑧 = 𝑍)
1615oveq1d 7417 . . . . . 6 ((πœ‘ ∧ 𝑧 = 𝑍) β†’ (𝑧𝐻𝑋) = (𝑍𝐻𝑋))
1714, 16eleqtrrd 2828 . . . . 5 ((πœ‘ ∧ 𝑧 = 𝑍) β†’ 𝐺 ∈ (𝑧𝐻𝑋))
18 moni.k . . . . . . . . 9 (πœ‘ β†’ 𝐾 ∈ (𝑍𝐻𝑋))
1918adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝑧 = 𝑍) β†’ 𝐾 ∈ (𝑍𝐻𝑋))
2019, 16eleqtrrd 2828 . . . . . . 7 ((πœ‘ ∧ 𝑧 = 𝑍) β†’ 𝐾 ∈ (𝑧𝐻𝑋))
2120adantr 480 . . . . . 6 (((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) β†’ 𝐾 ∈ (𝑧𝐻𝑋))
22 simpllr 773 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ 𝑧 = 𝑍)
2322opeq1d 4872 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ βŸ¨π‘§, π‘‹βŸ© = βŸ¨π‘, π‘‹βŸ©)
2423oveq1d 7417 . . . . . . . . 9 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ (βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ) = (βŸ¨π‘, π‘‹βŸ© Β· π‘Œ))
25 eqidd 2725 . . . . . . . . 9 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ 𝐹 = 𝐹)
26 simplr 766 . . . . . . . . 9 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ 𝑔 = 𝐺)
2724, 25, 26oveq123d 7423 . . . . . . . 8 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ (𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)𝑔) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐺))
28 simpr 484 . . . . . . . . 9 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ β„Ž = 𝐾)
2924, 25, 28oveq123d 7423 . . . . . . . 8 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ (𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)β„Ž) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐾))
3027, 29eqeq12d 2740 . . . . . . 7 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ ((𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)β„Ž) ↔ (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐺) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐾)))
3126, 28eqeq12d 2740 . . . . . . 7 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ (𝑔 = β„Ž ↔ 𝐺 = 𝐾))
3230, 31imbi12d 344 . . . . . 6 ((((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ β„Ž = 𝐾) β†’ (((𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)β„Ž) β†’ 𝑔 = β„Ž) ↔ ((𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐺) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐾) β†’ 𝐺 = 𝐾)))
3321, 32rspcdv 3596 . . . . 5 (((πœ‘ ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) β†’ (βˆ€β„Ž ∈ (𝑧𝐻𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)β„Ž) β†’ 𝑔 = β„Ž) β†’ ((𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐺) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐾) β†’ 𝐺 = 𝐾)))
3417, 33rspcimdv 3594 . . . 4 ((πœ‘ ∧ 𝑧 = 𝑍) β†’ (βˆ€π‘” ∈ (𝑧𝐻𝑋)βˆ€β„Ž ∈ (𝑧𝐻𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)β„Ž) β†’ 𝑔 = β„Ž) β†’ ((𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐺) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐾) β†’ 𝐺 = 𝐾)))
3512, 34rspcimdv 3594 . . 3 (πœ‘ β†’ (βˆ€π‘§ ∈ 𝐡 βˆ€π‘” ∈ (𝑧𝐻𝑋)βˆ€β„Ž ∈ (𝑧𝐻𝑋)((𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)𝑔) = (𝐹(βŸ¨π‘§, π‘‹βŸ© Β· π‘Œ)β„Ž) β†’ 𝑔 = β„Ž) β†’ ((𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐺) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐾) β†’ 𝐺 = 𝐾)))
3611, 35mpd 15 . 2 (πœ‘ β†’ ((𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐺) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐾) β†’ 𝐺 = 𝐾))
37 oveq2 7410 . 2 (𝐺 = 𝐾 β†’ (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐺) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐾))
3836, 37impbid1 224 1 (πœ‘ β†’ ((𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐺) = (𝐹(βŸ¨π‘, π‘‹βŸ© Β· π‘Œ)𝐾) ↔ 𝐺 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βŸ¨cop 4627  β€˜cfv 6534  (class class class)co 7402  Basecbs 17145  Hom chom 17209  compcco 17210  Catccat 17609  Monocmon 17676
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 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-cat 17613  df-mon 17678
This theorem is referenced by:  epii  17691  monsect  17731  fthmon  17881  setcmon  18041
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