funcoppc5 - Mathbox for Zhi Wang

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Theorem funcoppc5 49636

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 17824	. . . 4 ⊢ Rel (𝑂 Func 𝑃)
3		eqid 2737	. . . 4 ⊢ ( oppFunc ‘𝐹) = ( oppFunc ‘𝐹)
4	1, 2, 3	oppfrcl 49619	. . 3 ⊢ (𝜑 → 𝐹 ∈ (V × V))
5		1st2nd2 7976	. . 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 6840	. . . . . 6 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
12		df-ov 7365	. . . . . 6 ⊢ ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) = ( oppFunc ‘⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
13	11, 12	eqtr4di 2790	. . . . 5 ⊢ (𝜑 → ( oppFunc ‘𝐹) = ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)))
14	13, 1	eqeltrrd 2838	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹) oppFunc (2^nd ‘𝐹)) ∈ (𝑂 Func 𝑃))
15	7, 8, 9, 10, 14	funcoppc4 49635	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
16		df-br 5087	. . 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 2837	1 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ⟨cop 4574 class class class wbr 5086 × cxp 5624 ‘cfv 6494 (class class class)co 7362 1^st c1st 7935 2^nd c2nd 7936 oppCatcoppc 17672 Func cfunc 17816 oppFunc coppf 49613

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-er 8638 df-map 8770 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-hom 17239 df-cco 17240 df-cat 17629 df-cid 17630 df-homf 17631 df-comf 17632 df-oppc 17673 df-func 17820 df-oppf 49614

This theorem is referenced by: oppfuprcl 49695 natoppfb 49722 lmddu 50158 cmddu 50159

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