Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oppfval2 Structured version   Visualization version   GIF version

Theorem oppfval2 49864
Description: Value of the opposite functor. (Contributed by Zhi Wang, 13-Nov-2025.)
Assertion
Ref Expression
oppfval2 (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1st𝐹), tpos (2nd𝐹)⟩)

Proof of Theorem oppfval2
StepHypRef Expression
1 relfunc 17922 . . . . 5 Rel (𝐶 Func 𝐷)
2 1st2nd 8039 . . . . 5 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
31, 2mpan 702 . . . 4 (𝐹 ∈ (𝐶 Func 𝐷) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
43fveq2d 6889 . . 3 (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1st𝐹), (2nd𝐹)⟩))
5 df-ov 7417 . . 3 ((1st𝐹) oppFunc (2nd𝐹)) = ( oppFunc ‘⟨(1st𝐹), (2nd𝐹)⟩)
64, 5eqtr4di 2823 . 2 (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ((1st𝐹) oppFunc (2nd𝐹)))
7 1st2ndbr 8042 . . . 4 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
81, 7mpan 702 . . 3 (𝐹 ∈ (𝐶 Func 𝐷) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
9 oppfval 49863 . . 3 ((1st𝐹)(𝐶 Func 𝐷)(2nd𝐹) → ((1st𝐹) oppFunc (2nd𝐹)) = ⟨(1st𝐹), tpos (2nd𝐹)⟩)
108, 9syl 18 . 2 (𝐹 ∈ (𝐶 Func 𝐷) → ((1st𝐹) oppFunc (2nd𝐹)) = ⟨(1st𝐹), tpos (2nd𝐹)⟩)
116, 10eqtrd 2805 1 (𝐹 ∈ (𝐶 Func 𝐷) → ( oppFunc ‘𝐹) = ⟨(1st𝐹), tpos (2nd𝐹)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1568  wcel 2150  cop 4600   class class class wbr 5114  Rel wrel 5670  cfv 6540  (class class class)co 7414  1st c1st 7987  2nd c2nd 7988  tpos ctpos 8224   Func cfunc 17914   oppFunc coppf 49849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7989  df-2nd 7990  df-tpos 8225  df-map 8829  df-ixp 8899  df-func 17918  df-oppf 49850
This theorem is referenced by:  oppf1  49866  oppf2  49867  2oppffunc  49873  cofuoppf  49877  fulloppf  49890  fthoppf  49891  natoppf2  49957  opf11  50130  opf12  50131  ranval3  50358  islmd  50392
  Copyright terms: Public domain W3C validator