funcoppc5 - Mathbox for Zhi Wang

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Theorem funcoppc5 49140

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 17769	. . . 4 ⊢ Rel (𝑂 Func 𝑃)
3		eqid 2729	. . . 4 ⊢ ( oppFunc ‘𝐹) = ( oppFunc ‘𝐹)
4	1, 2, 3	oppfrcl 49123	. . 3 ⊢ (𝜑 → 𝐹 ∈ (V × V))
5		1st2nd2 7963	. . 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 6826	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
12		df-ov 7352	. . . . . 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 49139	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
16		df-br 5093	. . 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 3436 ⟨cop 4583 class class class wbr 5092 × cxp 5617 ‘cfv 6482 (class class class)co 7349 1^st c1st 7922 2^nd c2nd 7923 oppCatcoppc 17617 Func cfunc 17761 oppFunc coppf 49117

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-hom 17185 df-cco 17186 df-cat 17574 df-cid 17575 df-homf 17576 df-comf 17577 df-oppc 17618 df-func 17765 df-oppf 49118

This theorem is referenced by: oppfuprcl 49199 natoppfb 49226 lmddu 49662 cmddu 49663

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