oppfoppc2 - Mathbox for Zhi Wang

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Theorem oppfoppc2 49033

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 17873	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		oppfoppc2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		1st2nd 8036	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	1, 2, 3	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5	4	fveq2d 6879	. . 3 ⊢ (𝜑 → (oppFunc‘𝐹) = (oppFunc‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
6		df-ov 7406	. . 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 48991	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
11	8, 9, 10	oppfoppc 49032	. 2 ⊢ (𝜑 → ((1^st ‘𝐹)oppFunc(2^nd ‘𝐹)) ∈ (𝑂 Func 𝑃))
12	7, 11	eqeltrd 2834	1 ⊢ (𝜑 → (oppFunc‘𝐹) ∈ (𝑂 Func 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⟨cop 4607 Rel wrel 5659 ‘cfv 6530 (class class class)co 7403 1^st c1st 7984 2^nd c2nd 7985 oppCatcoppc 17721 Func cfunc 17865 oppFunccoppf 49019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8717 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-hom 17293 df-cco 17294 df-cat 17678 df-cid 17679 df-oppc 17722 df-func 17869 df-oppf 49020

This theorem is referenced by: (None)

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