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Theorem cofidf2 49750
Description: If "𝐹 is a section of 𝐺 " in a category of small categories (in a universe), then the morphism part of 𝐹 is injective, and the morphism part of 𝐺 is surjective in the image of 𝐹. (Contributed by Zhi Wang, 15-Nov-2025.)
Hypotheses
Ref Expression
cofidval.i 𝐼 = (idfunc𝐷)
cofidval.b 𝐵 = (Base‘𝐷)
cofidval.f (𝜑𝐹(𝐷 Func 𝐸)𝐺)
cofidval.k (𝜑𝐾(𝐸 Func 𝐷)𝐿)
cofidval.o (𝜑 → (⟨𝐾, 𝐿⟩ ∘func𝐹, 𝐺⟩) = 𝐼)
cofidval.h 𝐻 = (Hom ‘𝐷)
cofidf2.j 𝐽 = (Hom ‘𝐸)
cofidf2.x (𝜑𝑋𝐵)
cofidf2.y (𝜑𝑌𝐵)
Assertion
Ref Expression
cofidf2 (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)) ∧ ((𝐹𝑋)𝐿(𝐹𝑌)):((𝐹𝑋)𝐽(𝐹𝑌))–onto→(𝑋𝐻𝑌)))

Proof of Theorem cofidf2
StepHypRef Expression
1 cofidval.i . . 3 𝐼 = (idfunc𝐷)
2 cofidval.b . . 3 𝐵 = (Base‘𝐷)
3 cofidval.f . . . 4 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
4 df-br 5105 . . . 4 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
53, 4sylib 221 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
6 cofidval.k . . . 4 (𝜑𝐾(𝐸 Func 𝐷)𝐿)
7 df-br 5105 . . . 4 (𝐾(𝐸 Func 𝐷)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐸 Func 𝐷))
86, 7sylib 221 . . 3 (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐸 Func 𝐷))
9 cofidval.o . . 3 (𝜑 → (⟨𝐾, 𝐿⟩ ∘func𝐹, 𝐺⟩) = 𝐼)
10 cofidval.h . . 3 𝐻 = (Hom ‘𝐷)
11 cofidf2.j . . 3 𝐽 = (Hom ‘𝐸)
12 cofidf2.x . . 3 (𝜑𝑋𝐵)
13 cofidf2.y . . 3 (𝜑𝑌𝐵)
141, 2, 5, 8, 9, 10, 11, 12, 13cofidf2a 49747 . 2 (𝜑 → ((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) ∧ (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)):(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌))–onto→(𝑋𝐻𝑌)))
153func2nd 49708 . . . . 5 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
1615oveqd 7417 . . . 4 (𝜑 → (𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌) = (𝑋𝐺𝑌))
17 eqidd 2766 . . . 4 (𝜑 → (𝑋𝐻𝑌) = (𝑋𝐻𝑌))
183func1st 49707 . . . . . 6 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
1918fveq1d 6873 . . . . 5 (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑋) = (𝐹𝑋))
2018fveq1d 6873 . . . . 5 (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑌) = (𝐹𝑌))
2119, 20oveq12d 7418 . . . 4 (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹𝑋)𝐽(𝐹𝑌)))
2216, 17, 21f1eq123d 6802 . . 3 (𝜑 → ((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) ↔ (𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌))))
236func2nd 49708 . . . . 5 (𝜑 → (2nd ‘⟨𝐾, 𝐿⟩) = 𝐿)
2423, 19, 20oveq123d 7421 . . . 4 (𝜑 → (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) = ((𝐹𝑋)𝐿(𝐹𝑌)))
2524, 21, 17foeq123d 6803 . . 3 (𝜑 → ((((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)):(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌))–onto→(𝑋𝐻𝑌) ↔ ((𝐹𝑋)𝐿(𝐹𝑌)):((𝐹𝑋)𝐽(𝐹𝑌))–onto→(𝑋𝐻𝑌)))
2622, 25anbi12d 643 . 2 (𝜑 → (((𝑋(2nd ‘⟨𝐹, 𝐺⟩)𝑌):(𝑋𝐻𝑌)–1-1→(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)) ∧ (((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)(2nd ‘⟨𝐾, 𝐿⟩)((1st ‘⟨𝐹, 𝐺⟩)‘𝑌)):(((1st ‘⟨𝐹, 𝐺⟩)‘𝑋)𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑌))–onto→(𝑋𝐻𝑌)) ↔ ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)) ∧ ((𝐹𝑋)𝐿(𝐹𝑌)):((𝐹𝑋)𝐽(𝐹𝑌))–onto→(𝑋𝐻𝑌))))
2714, 26mpbid 235 1 (𝜑 → ((𝑋𝐺𝑌):(𝑋𝐻𝑌)–1-1→((𝐹𝑋)𝐽(𝐹𝑌)) ∧ ((𝐹𝑋)𝐿(𝐹𝑌)):((𝐹𝑋)𝐽(𝐹𝑌))–onto→(𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 400   = wceq 1563  wcel 2145  cop 4591   class class class wbr 5104  1-1wf1 6522  ontowfo 6523  cfv 6525  (class class class)co 7400  1st c1st 7972  2nd c2nd 7973  Basecbs 17257  Hom chom 17309   Func cfunc 17899  idfunccidfu 17900  func ccofu 17901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-func 17903  df-idfu 17904  df-cofu 17905
This theorem is referenced by:  cofidfth  49792
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