funcoppc5 - Mathbox for Zhi Wang

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Theorem funcoppc5 49138

Description: A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025.)

**Hypotheses**
Ref	Expression
funcoppc2.o	⊢ 𝑂 = (oppCat‘𝐶)
funcoppc2.p	⊢ 𝑃 = (oppCat‘𝐷)
funcoppc2.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
funcoppc2.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)
funcoppc5.f	⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))

**Assertion**
Ref	Expression
funcoppc5	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

**Proof of Theorem funcoppc5**
Step	Hyp	Ref	Expression
1		funcoppc5.f	. . . 4 ⊢ (𝜑 → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))
2		relfunc 17831	. . . 4 ⊢ Rel (𝑂 Func 𝑃)
3		eqid 2730	. . . 4 ⊢ ( oppFunc ‘𝐹) = ( oppFunc ‘𝐹)
4	1, 2, 3	oppfrcl 49121	. . 3 ⊢ (𝜑 → 𝐹 ∈ (V × V))
5		1st2nd2 8010	. . 3 ⊢ (𝐹 ∈ (V × V) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
7		funcoppc2.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
8		funcoppc2.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐷)
9		funcoppc2.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
10		funcoppc2.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
11	6	fveq2d 6865	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
12		df-ov 7393	. . . . . 6 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
13	11, 12	eqtr4di 2783	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)))
14	13, 1	eqeltrrd 2830	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) ∈ (𝑂 Func 𝑃))
15	7, 8, 9, 10, 14	funcoppc4 49137	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
16		df-br 5111	. . 3 ⊢ ((1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹) ↔ ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩ ∈ (𝐶 Func 𝐷))
17	15, 16	sylib 218	. 2 ⊢ (𝜑 → ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩ ∈ (𝐶 Func 𝐷))
18	6, 17	eqeltrd 2829	1 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ⟨cop 4598 class class class wbr 5110 × cxp 5639 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 oppCatcoppc 17679 Func cfunc 17823 oppFunc coppf 49115

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2066 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 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-hom 17251 df-cco 17252 df-cat 17636 df-cid 17637 df-homf 17638 df-comf 17639 df-oppc 17680 df-func 17827 df-oppf 49116

This theorem is referenced by: oppfuprcl 49197 natoppfb 49224 lmddu 49660 cmddu 49661

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