cofuswapf1 - Mathbox for Zhi Wang

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Theorem cofuswapf1 48967

Description: The object part of a bifunctor pre-composed with a swap functor. (Contributed by Zhi Wang, 9-Oct-2025.)

**Hypotheses**
Ref	Expression
cofuswapf1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
cofuswapf1.d	⊢ (𝜑 → 𝐷 ∈ Cat)
cofuswapf1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×_c 𝐶) Func 𝐸))
cofuswapf1.g	⊢ (𝜑 → 𝐺 = (𝐹 ∘_func (𝐶swap_F𝐷)))
cofuswapf1.a	⊢ 𝐴 = (Base‘𝐶)
cofuswapf1.b	⊢ 𝐵 = (Base‘𝐷)
cofuswapf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
cofuswapf1.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
cofuswapf1	⊢ (𝜑 → (𝑋(1^st ‘𝐺)𝑌) = (𝑌(1^st ‘𝐹)𝑋))

**Proof of Theorem cofuswapf1**
Step	Hyp	Ref	Expression
1		df-ov 7432	. . . 4 ⊢ (𝑋(1^st ‘𝐺)𝑌) = ((1^st ‘𝐺)‘⟨𝑋, 𝑌⟩)
2		cofuswapf1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝐹 ∘_func (𝐶swap_F𝐷)))
3	2	fveq2d 6908	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺) = (1^st ‘(𝐹 ∘_func (𝐶swap_F𝐷))))
4	3	fveq1d 6906	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐺)‘⟨𝑋, 𝑌⟩) = ((1^st ‘(𝐹 ∘_func (𝐶swap_F𝐷)))‘⟨𝑋, 𝑌⟩))
5	1, 4	eqtrid 2788	. . 3 ⊢ (𝜑 → (𝑋(1^st ‘𝐺)𝑌) = ((1^st ‘(𝐹 ∘_func (𝐶swap_F𝐷)))‘⟨𝑋, 𝑌⟩))
6		eqid 2736	. . . . 5 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
7		cofuswapf1.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
8		cofuswapf1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
9	6, 7, 8	xpcbas 18219	. . . 4 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
10		cofuswapf1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		cofuswapf1.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
12		eqid 2736	. . . . 5 ⊢ (𝐷 ×_c 𝐶) = (𝐷 ×_c 𝐶)
13	10, 11, 6, 12	swapffunca 48963	. . . 4 ⊢ (𝜑 → (𝐶swap_F𝐷) ∈ ((𝐶 ×_c 𝐷) Func (𝐷 ×_c 𝐶)))
14		cofuswapf1.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×_c 𝐶) Func 𝐸))
15		cofuswapf1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
16		cofuswapf1.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
17	15, 16	opelxpd 5722	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
18	9, 13, 14, 17	cofu1 17925	. . 3 ⊢ (𝜑 → ((1^st ‘(𝐹 ∘_func (𝐶swap_F𝐷)))‘⟨𝑋, 𝑌⟩) = ((1^st ‘𝐹)‘((1^st ‘(𝐶swap_F𝐷))‘⟨𝑋, 𝑌⟩)))
19		df-ov 7432	. . . . 5 ⊢ (𝑋(1^st ‘(𝐶swap_F𝐷))𝑌) = ((1^st ‘(𝐶swap_F𝐷))‘⟨𝑋, 𝑌⟩)
20	10, 11	swapfelvv 48942	. . . . . . 7 ⊢ (𝜑 → (𝐶swap_F𝐷) ∈ (V × V))
21		1st2nd2 8049	. . . . . . 7 ⊢ ((𝐶swap_F𝐷) ∈ (V × V) → (𝐶swap_F𝐷) = ⟨(1^st ‘(𝐶swap_F𝐷)), (2^nd ‘(𝐶swap_F𝐷))⟩)
22	20, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐶swap_F𝐷) = ⟨(1^st ‘(𝐶swap_F𝐷)), (2^nd ‘(𝐶swap_F𝐷))⟩)
23	15, 7	eleqtrdi 2850	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
24	16, 8	eleqtrdi 2850	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
25	22, 23, 24	swapf1 48951	. . . . 5 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶swap_F𝐷))𝑌) = ⟨𝑌, 𝑋⟩)
26	19, 25	eqtr3id 2790	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐶swap_F𝐷))‘⟨𝑋, 𝑌⟩) = ⟨𝑌, 𝑋⟩)
27	26	fveq2d 6908	. . 3 ⊢ (𝜑 → ((1^st ‘𝐹)‘((1^st ‘(𝐶swap_F𝐷))‘⟨𝑋, 𝑌⟩)) = ((1^st ‘𝐹)‘⟨𝑌, 𝑋⟩))
28	5, 18, 27	3eqtrd 2780	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝐺)𝑌) = ((1^st ‘𝐹)‘⟨𝑌, 𝑋⟩))
29		df-ov 7432	. 2 ⊢ (𝑌(1^st ‘𝐹)𝑋) = ((1^st ‘𝐹)‘⟨𝑌, 𝑋⟩)
30	28, 29	eqtr4di 2794	1 ⊢ (𝜑 → (𝑋(1^st ‘𝐺)𝑌) = (𝑌(1^st ‘𝐹)𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3479 ⟨cop 4630 × cxp 5681 ‘cfv 6559 (class class class)co 7429 1^st c1st 8008 2^nd c2nd 8009 Basecbs 17243 Catccat 17703 Func cfunc 17895 ∘_func ccofu 17897 ×_c cxpc 18209 swap_Fcswapf 48938

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-map 8864 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17244 df-hom 17317 df-cco 17318 df-cat 17707 df-cid 17708 df-func 17899 df-cofu 17901 df-xpc 18213 df-swapf 48939

This theorem is referenced by: tposcurf11 48970

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