oppfoppc2 - Mathbox for Zhi Wang

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Theorem oppfoppc2 49605

Description: The opposite functor is a functor on opposite categories. (Contributed by Zhi Wang, 14-Nov-2025.)

**Hypotheses**
Ref	Expression
oppfoppc.o	⊢ 𝑂 = (oppCat‘𝐶)
oppfoppc.p	⊢ 𝑃 = (oppCat‘𝐷)
oppfoppc2.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
oppfoppc2	⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))

**Proof of Theorem oppfoppc2**
Step	Hyp	Ref	Expression
1		relfunc 17818	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		oppfoppc2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		1st2nd 7981	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	1, 2, 3	sylancr 588	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5	4	fveq2d 6833	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
6		df-ov 7359	. . 3 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
7	5, 6	eqtr4di 2788	. 2 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)))
8		oppfoppc.o	. . 3 ⊢ 𝑂 = (oppCat‘𝐶)
9		oppfoppc.p	. . 3 ⊢ 𝑃 = (oppCat‘𝐷)
10	2	func1st2nd 49539	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
11	8, 9, 10	oppfoppc 49604	. 2 ⊢ (𝜑 → ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) ∈ (𝑂 Func 𝑃))
12	7, 11	eqeltrd 2835	1 ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4563 Rel wrel 5625 ‘cfv 6487 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 oppCatcoppc 17666 Func cfunc 17810 oppFunc coppf 49585

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8632 df-map 8764 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-hom 17233 df-cco 17234 df-cat 17623 df-cid 17624 df-oppc 17667 df-func 17814 df-oppf 49586

This theorem is referenced by: oppff1 49611 oppfdiag1 49877 oppfdiag 49879 lmddu 50130 cmddu 50131 lmdran 50134

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