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 49153

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 17761	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		oppfoppc2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		1st2nd 7966	. . . . 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 6821	. . 3 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
6		df-ov 7344	. . 3 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
7	5, 6	eqtr4di 2783	. 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 49087	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
11	8, 9, 10	oppfoppc 49152	. 2 ⊢ (𝜑 → ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) ∈ (𝑂 Func 𝑃))
12	7, 11	eqeltrd 2829	1 ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2110 ⟨cop 4580 Rel wrel 5619 ‘cfv 6477 (class class class)co 7341 1^st c1st 7914 2^nd c2nd 7915 oppCatcoppc 17609 Func cfunc 17753 oppFunc coppf 49133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-map 8747 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-hom 17177 df-cco 17178 df-cat 17566 df-cid 17567 df-oppc 17610 df-func 17757 df-oppf 49134

This theorem is referenced by: oppff1 49159 oppfdiag1 49425 oppfdiag 49427 lmddu 49678 cmddu 49679 lmdran 49682

W3C validator