funcoppc5 - Mathbox for Zhi Wang

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Theorem funcoppc5 49127

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 17804	. . . 4 ⊢ Rel (𝑂 Func 𝑃)
3		eqid 2729	. . . 4 ⊢ ( oppFunc ‘𝐹) = ( oppFunc ‘𝐹)
4	1, 2, 3	oppfrcl 49110	. . 3 ⊢ (𝜑 → 𝐹 ∈ (V × V))
5		1st2nd2 7986	. . 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 6844	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
12		df-ov 7372	. . . . . 6 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
13	11, 12	eqtr4di 2782	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)))
14	13, 1	eqeltrrd 2829	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) ∈ (𝑂 Func 𝑃))
15	7, 8, 9, 10, 14	funcoppc4 49126	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
16		df-br 5103	. . 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 2828	1 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 Vcvv 3444 ⟨cop 4591 class class class wbr 5102 × cxp 5629 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 oppCatcoppc 17652 Func cfunc 17796 oppFunc coppf 49104

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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-hom 17220 df-cco 17221 df-cat 17609 df-cid 17610 df-homf 17611 df-comf 17612 df-oppc 17653 df-func 17800 df-oppf 49105

This theorem is referenced by: oppfuprcl 49186 natoppfb 49213 lmddu 49649 cmddu 49650

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