oppfoppc2 - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > oppfoppc2		Structured version Visualization version GIF version

Theorem oppfoppc2 49754

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 17905	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		oppfoppc2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		1st2nd 8020	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	1, 2, 3	sylancr 596	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5	4	fveq2d 6871	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
6		df-ov 7399	. . 3 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
7	5, 6	eqtr4di 2816	. 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 49688	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
11	8, 9, 10	oppfoppc 49753	. 2 ⊢ (𝜑 → ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) ∈ (𝑂 Func 𝑃))
12	7, 11	eqeltrd 2863	1 ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1561 ∈ wcel 2143 ⟨cop 4589 Rel wrel 5653 ‘cfv 6521 (class class class)co 7396 1^st c1st 7968 2^nd c2nd 7969 oppCatcoppc 17753 Func cfunc 17897 oppFunc coppf 49734

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-er 8678 df-map 8810 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-nn 12221 df-2 12290 df-3 12291 df-4 12292 df-5 12293 df-6 12294 df-7 12295 df-8 12296 df-9 12297 df-n0 12492 df-z 12579 df-dec 12699 df-sets 17210 df-slot 17228 df-ndx 17240 df-base 17256 df-hom 17320 df-cco 17321 df-cat 17710 df-cid 17711 df-oppc 17754 df-func 17901 df-oppf 49735

This theorem is referenced by: oppff1 49760 oppfdiag1 50026 oppfdiag 50028 lmddu 50279 cmddu 50280 lmdran 50283

W3C validator