cofuswapf1 - Mathbox for Zhi Wang

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Theorem cofuswapf1 49039

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 7416	. . . 4 ⊢ (𝑋(1^st ‘𝐺)𝑌) = ((1^st ‘𝐺)‘⟨𝑋, 𝑌⟩)
2		cofuswapf1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝐹 ∘_func (𝐶swap_F𝐷)))
3	2	fveq2d 6890	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐺) = (1^st ‘(𝐹 ∘_func (𝐶swap_F𝐷))))
4	3	fveq1d 6888	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐺)‘⟨𝑋, 𝑌⟩) = ((1^st ‘(𝐹 ∘_func (𝐶swap_F𝐷)))‘⟨𝑋, 𝑌⟩))
5	1, 4	eqtrid 2781	. . 3 ⊢ (𝜑 → (𝑋(1^st ‘𝐺)𝑌) = ((1^st ‘(𝐹 ∘_func (𝐶swap_F𝐷)))‘⟨𝑋, 𝑌⟩))
6		eqid 2734	. . . . 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 18194	. . . 4 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×_c 𝐷))
10		cofuswapf1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		cofuswapf1.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
12		eqid 2734	. . . . 5 ⊢ (𝐷 ×_c 𝐶) = (𝐷 ×_c 𝐶)
13	10, 11, 6, 12	swapffunca 49035	. . . 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 5704	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
18	9, 13, 14, 17	cofu1 17901	. . 3 ⊢ (𝜑 → ((1^st ‘(𝐹 ∘_func (𝐶swap_F𝐷)))‘⟨𝑋, 𝑌⟩) = ((1^st ‘𝐹)‘((1^st ‘(𝐶swap_F𝐷))‘⟨𝑋, 𝑌⟩)))
19		df-ov 7416	. . . . 5 ⊢ (𝑋(1^st ‘(𝐶swap_F𝐷))𝑌) = ((1^st ‘(𝐶swap_F𝐷))‘⟨𝑋, 𝑌⟩)
20	10, 11	swapfelvv 49014	. . . . . . 7 ⊢ (𝜑 → (𝐶swap_F𝐷) ∈ (V × V))
21		1st2nd2 8035	. . . . . . 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 2843	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
24	16, 8	eleqtrdi 2843	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
25	22, 23, 24	swapf1 49023	. . . . 5 ⊢ (𝜑 → (𝑋(1^st ‘(𝐶swap_F𝐷))𝑌) = ⟨𝑌, 𝑋⟩)
26	19, 25	eqtr3id 2783	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐶swap_F𝐷))‘⟨𝑋, 𝑌⟩) = ⟨𝑌, 𝑋⟩)
27	26	fveq2d 6890	. . 3 ⊢ (𝜑 → ((1^st ‘𝐹)‘((1^st ‘(𝐶swap_F𝐷))‘⟨𝑋, 𝑌⟩)) = ((1^st ‘𝐹)‘⟨𝑌, 𝑋⟩))
28	5, 18, 27	3eqtrd 2773	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝐺)𝑌) = ((1^st ‘𝐹)‘⟨𝑌, 𝑋⟩))
29		df-ov 7416	. 2 ⊢ (𝑌(1^st ‘𝐹)𝑋) = ((1^st ‘𝐹)‘⟨𝑌, 𝑋⟩)
30	28, 29	eqtr4di 2787	1 ⊢ (𝜑 → (𝑋(1^st ‘𝐺)𝑌) = (𝑌(1^st ‘𝐹)𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3463 ⟨cop 4612 × cxp 5663 ‘cfv 6541 (class class class)co 7413 1^st c1st 7994 2^nd c2nd 7995 Basecbs 17230 Catccat 17679 Func cfunc 17871 ∘_func ccofu 17873 ×_c cxpc 18184 swap_Fcswapf 49010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8727 df-map 8850 df-ixp 8920 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12510 df-z 12597 df-dec 12717 df-uz 12861 df-fz 13530 df-struct 17167 df-slot 17202 df-ndx 17214 df-base 17231 df-hom 17298 df-cco 17299 df-cat 17683 df-cid 17684 df-func 17875 df-cofu 17877 df-xpc 18188 df-swapf 49011

This theorem is referenced by: tposcurf11 49042

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