swapfcoa - Mathbox for Zhi Wang

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Theorem swapfcoa 49243

Description: Composition in the source category is mapped to composition in the target. (𝜑 → 𝐶 ∈ Cat) and (𝜑 → 𝐷 ∈ Cat) can be replaced by a weaker hypothesis (𝜑 → 𝑆 ∈ Cat). (Contributed by Zhi Wang, 8-Oct-2025.)

**Hypotheses**
Ref	Expression
swapfid.c	⊢ (𝜑 → 𝐶 ∈ Cat)
swapfid.d	⊢ (𝜑 → 𝐷 ∈ Cat)
swapfid.s	⊢ 𝑆 = (𝐶 ×_c 𝐷)
swapfid.t	⊢ 𝑇 = (𝐷 ×_c 𝐶)
swapfid.o	⊢ (𝜑 → (𝐶 swap_F 𝐷) = ⟨𝑂, 𝑃⟩)
swapfida.b	⊢ 𝐵 = (Base‘𝑆)
swapfida.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
swapfcoa.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
swapfcoa.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
swapfcoa.h	⊢ 𝐻 = (Hom ‘𝑆)
swapfcoa.m	⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
swapfcoa.n	⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑍))
swapfcoa.os	⊢ · = (comp‘𝑆)
swapfcoa.ot	⊢ ∙ = (comp‘𝑇)

**Assertion**
Ref	Expression
swapfcoa	⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝑃𝑍)‘𝑁)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝑀)))

**Proof of Theorem swapfcoa**
Step	Hyp	Ref	Expression
1		swapfid.o	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 swap_F 𝐷) = ⟨𝑂, 𝑃⟩)
2		swapfid.s	. . . . . . . . 9 ⊢ 𝑆 = (𝐶 ×_c 𝐷)
3		swapfida.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑆)
4		swapfida.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5	1, 2, 3, 4	swapf1a 49231	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩)
6	5	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (1^st ‘(𝑂‘𝑋)) = (1^st ‘⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩))
7		fvex 6853	. . . . . . . 8 ⊢ (2^nd ‘𝑋) ∈ V
8		fvex 6853	. . . . . . . 8 ⊢ (1^st ‘𝑋) ∈ V
9	7, 8	op1st 7955	. . . . . . 7 ⊢ (1^st ‘⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩) = (2^nd ‘𝑋)
10	6, 9	eqtrdi 2780	. . . . . 6 ⊢ (𝜑 → (1^st ‘(𝑂‘𝑋)) = (2^nd ‘𝑋))
11		swapfcoa.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
12	1, 2, 3, 11	swapf1a 49231	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩)
13	12	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (1^st ‘(𝑂‘𝑌)) = (1^st ‘⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩))
14		fvex 6853	. . . . . . . 8 ⊢ (2^nd ‘𝑌) ∈ V
15		fvex 6853	. . . . . . . 8 ⊢ (1^st ‘𝑌) ∈ V
16	14, 15	op1st 7955	. . . . . . 7 ⊢ (1^st ‘⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩) = (2^nd ‘𝑌)
17	13, 16	eqtrdi 2780	. . . . . 6 ⊢ (𝜑 → (1^st ‘(𝑂‘𝑌)) = (2^nd ‘𝑌))
18	10, 17	opeq12d 4841	. . . . 5 ⊢ (𝜑 → ⟨(1^st ‘(𝑂‘𝑋)), (1^st ‘(𝑂‘𝑌))⟩ = ⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩)
19		swapfcoa.z	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
20	1, 2, 3, 19	swapf1a 49231	. . . . . . 7 ⊢ (𝜑 → (𝑂‘𝑍) = ⟨(2^nd ‘𝑍), (1^st ‘𝑍)⟩)
21	20	fveq2d 6844	. . . . . 6 ⊢ (𝜑 → (1^st ‘(𝑂‘𝑍)) = (1^st ‘⟨(2^nd ‘𝑍), (1^st ‘𝑍)⟩))
22		fvex 6853	. . . . . . 7 ⊢ (2^nd ‘𝑍) ∈ V
23		fvex 6853	. . . . . . 7 ⊢ (1^st ‘𝑍) ∈ V
24	22, 23	op1st 7955	. . . . . 6 ⊢ (1^st ‘⟨(2^nd ‘𝑍), (1^st ‘𝑍)⟩) = (2^nd ‘𝑍)
25	21, 24	eqtrdi 2780	. . . . 5 ⊢ (𝜑 → (1^st ‘(𝑂‘𝑍)) = (2^nd ‘𝑍))
26	18, 25	oveq12d 7387	. . . 4 ⊢ (𝜑 → (⟨(1^st ‘(𝑂‘𝑋)), (1^st ‘(𝑂‘𝑌))⟩(comp‘𝐷)(1^st ‘(𝑂‘𝑍))) = (⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍)))
27		swapfcoa.h	. . . . . . . 8 ⊢ 𝐻 = (Hom ‘𝑆)
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
29		swapfcoa.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑍))
30	1, 2, 3, 11, 19, 28, 29	swapf2a 49233	. . . . . 6 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝑁) = ⟨(2^nd ‘𝑁), (1^st ‘𝑁)⟩)
31	30	fveq2d 6844	. . . . 5 ⊢ (𝜑 → (1^st ‘((𝑌𝑃𝑍)‘𝑁)) = (1^st ‘⟨(2^nd ‘𝑁), (1^st ‘𝑁)⟩))
32		fvex 6853	. . . . . 6 ⊢ (2^nd ‘𝑁) ∈ V
33		fvex 6853	. . . . . 6 ⊢ (1^st ‘𝑁) ∈ V
34	32, 33	op1st 7955	. . . . 5 ⊢ (1^st ‘⟨(2^nd ‘𝑁), (1^st ‘𝑁)⟩) = (2^nd ‘𝑁)
35	31, 34	eqtrdi 2780	. . . 4 ⊢ (𝜑 → (1^st ‘((𝑌𝑃𝑍)‘𝑁)) = (2^nd ‘𝑁))
36		swapfcoa.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
37	1, 2, 3, 4, 11, 28, 36	swapf2a 49233	. . . . . 6 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝑀) = ⟨(2^nd ‘𝑀), (1^st ‘𝑀)⟩)
38	37	fveq2d 6844	. . . . 5 ⊢ (𝜑 → (1^st ‘((𝑋𝑃𝑌)‘𝑀)) = (1^st ‘⟨(2^nd ‘𝑀), (1^st ‘𝑀)⟩))
39		fvex 6853	. . . . . 6 ⊢ (2^nd ‘𝑀) ∈ V
40		fvex 6853	. . . . . 6 ⊢ (1^st ‘𝑀) ∈ V
41	39, 40	op1st 7955	. . . . 5 ⊢ (1^st ‘⟨(2^nd ‘𝑀), (1^st ‘𝑀)⟩) = (2^nd ‘𝑀)
42	38, 41	eqtrdi 2780	. . . 4 ⊢ (𝜑 → (1^st ‘((𝑋𝑃𝑌)‘𝑀)) = (2^nd ‘𝑀))
43	26, 35, 42	oveq123d 7390	. . 3 ⊢ (𝜑 → ((1^st ‘((𝑌𝑃𝑍)‘𝑁))(⟨(1^st ‘(𝑂‘𝑋)), (1^st ‘(𝑂‘𝑌))⟩(comp‘𝐷)(1^st ‘(𝑂‘𝑍)))(1^st ‘((𝑋𝑃𝑌)‘𝑀))) = ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀)))
44	5	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘(𝑂‘𝑋)) = (2^nd ‘⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩))
45	7, 8	op2nd 7956	. . . . . . 7 ⊢ (2^nd ‘⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩) = (1^st ‘𝑋)
46	44, 45	eqtrdi 2780	. . . . . 6 ⊢ (𝜑 → (2^nd ‘(𝑂‘𝑋)) = (1^st ‘𝑋))
47	12	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘(𝑂‘𝑌)) = (2^nd ‘⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩))
48	14, 15	op2nd 7956	. . . . . . 7 ⊢ (2^nd ‘⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩) = (1^st ‘𝑌)
49	47, 48	eqtrdi 2780	. . . . . 6 ⊢ (𝜑 → (2^nd ‘(𝑂‘𝑌)) = (1^st ‘𝑌))
50	46, 49	opeq12d 4841	. . . . 5 ⊢ (𝜑 → ⟨(2^nd ‘(𝑂‘𝑋)), (2^nd ‘(𝑂‘𝑌))⟩ = ⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩)
51	20	fveq2d 6844	. . . . . 6 ⊢ (𝜑 → (2^nd ‘(𝑂‘𝑍)) = (2^nd ‘⟨(2^nd ‘𝑍), (1^st ‘𝑍)⟩))
52	22, 23	op2nd 7956	. . . . . 6 ⊢ (2^nd ‘⟨(2^nd ‘𝑍), (1^st ‘𝑍)⟩) = (1^st ‘𝑍)
53	51, 52	eqtrdi 2780	. . . . 5 ⊢ (𝜑 → (2^nd ‘(𝑂‘𝑍)) = (1^st ‘𝑍))
54	50, 53	oveq12d 7387	. . . 4 ⊢ (𝜑 → (⟨(2^nd ‘(𝑂‘𝑋)), (2^nd ‘(𝑂‘𝑌))⟩(comp‘𝐶)(2^nd ‘(𝑂‘𝑍))) = (⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍)))
55	30	fveq2d 6844	. . . . 5 ⊢ (𝜑 → (2^nd ‘((𝑌𝑃𝑍)‘𝑁)) = (2^nd ‘⟨(2^nd ‘𝑁), (1^st ‘𝑁)⟩))
56	32, 33	op2nd 7956	. . . . 5 ⊢ (2^nd ‘⟨(2^nd ‘𝑁), (1^st ‘𝑁)⟩) = (1^st ‘𝑁)
57	55, 56	eqtrdi 2780	. . . 4 ⊢ (𝜑 → (2^nd ‘((𝑌𝑃𝑍)‘𝑁)) = (1^st ‘𝑁))
58	37	fveq2d 6844	. . . . 5 ⊢ (𝜑 → (2^nd ‘((𝑋𝑃𝑌)‘𝑀)) = (2^nd ‘⟨(2^nd ‘𝑀), (1^st ‘𝑀)⟩))
59	39, 40	op2nd 7956	. . . . 5 ⊢ (2^nd ‘⟨(2^nd ‘𝑀), (1^st ‘𝑀)⟩) = (1^st ‘𝑀)
60	58, 59	eqtrdi 2780	. . . 4 ⊢ (𝜑 → (2^nd ‘((𝑋𝑃𝑌)‘𝑀)) = (1^st ‘𝑀))
61	54, 57, 60	oveq123d 7390	. . 3 ⊢ (𝜑 → ((2^nd ‘((𝑌𝑃𝑍)‘𝑁))(⟨(2^nd ‘(𝑂‘𝑋)), (2^nd ‘(𝑂‘𝑌))⟩(comp‘𝐶)(2^nd ‘(𝑂‘𝑍)))(2^nd ‘((𝑋𝑃𝑌)‘𝑀))) = ((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)))
62	43, 61	opeq12d 4841	. 2 ⊢ (𝜑 → ⟨((1^st ‘((𝑌𝑃𝑍)‘𝑁))(⟨(1^st ‘(𝑂‘𝑋)), (1^st ‘(𝑂‘𝑌))⟩(comp‘𝐷)(1^st ‘(𝑂‘𝑍)))(1^st ‘((𝑋𝑃𝑌)‘𝑀))), ((2^nd ‘((𝑌𝑃𝑍)‘𝑁))(⟨(2^nd ‘(𝑂‘𝑋)), (2^nd ‘(𝑂‘𝑌))⟩(comp‘𝐶)(2^nd ‘(𝑂‘𝑍)))(2^nd ‘((𝑋𝑃𝑌)‘𝑀)))⟩ = ⟨((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀)), ((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀))⟩)
63		swapfid.t	. . 3 ⊢ 𝑇 = (𝐷 ×_c 𝐶)
64		eqid 2729	. . 3 ⊢ (Base‘𝑇) = (Base‘𝑇)
65		eqid 2729	. . 3 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
66		eqid 2729	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
67		eqid 2729	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
68		swapfcoa.ot	. . 3 ⊢ ∙ = (comp‘𝑇)
69		swapfid.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
70		swapfid.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
71	1, 2, 63, 69, 70, 3, 64	swapf1f1o 49237	. . . . 5 ⊢ (𝜑 → 𝑂:𝐵–1-1-onto→(Base‘𝑇))
72		f1of 6782	. . . . 5 ⊢ (𝑂:𝐵–1-1-onto→(Base‘𝑇) → 𝑂:𝐵⟶(Base‘𝑇))
73	71, 72	syl 17	. . . 4 ⊢ (𝜑 → 𝑂:𝐵⟶(Base‘𝑇))
74	73, 4	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (Base‘𝑇))
75	73, 11	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝑂‘𝑌) ∈ (Base‘𝑇))
76	73, 19	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝑂‘𝑍) ∈ (Base‘𝑇))
77	1, 2, 63, 27, 65, 3, 4, 11	swapf2f1oa 49239	. . . . 5 ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)(Hom ‘𝑇)(𝑂‘𝑌)))
78		f1of 6782	. . . . 5 ⊢ ((𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)(Hom ‘𝑇)(𝑂‘𝑌)) → (𝑋𝑃𝑌):(𝑋𝐻𝑌)⟶((𝑂‘𝑋)(Hom ‘𝑇)(𝑂‘𝑌)))
79	77, 78	syl 17	. . . 4 ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)⟶((𝑂‘𝑋)(Hom ‘𝑇)(𝑂‘𝑌)))
80	79, 36	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝑀) ∈ ((𝑂‘𝑋)(Hom ‘𝑇)(𝑂‘𝑌)))
81	1, 2, 63, 27, 65, 3, 11, 19	swapf2f1oa 49239	. . . . 5 ⊢ (𝜑 → (𝑌𝑃𝑍):(𝑌𝐻𝑍)–1-1-onto→((𝑂‘𝑌)(Hom ‘𝑇)(𝑂‘𝑍)))
82		f1of 6782	. . . . 5 ⊢ ((𝑌𝑃𝑍):(𝑌𝐻𝑍)–1-1-onto→((𝑂‘𝑌)(Hom ‘𝑇)(𝑂‘𝑍)) → (𝑌𝑃𝑍):(𝑌𝐻𝑍)⟶((𝑂‘𝑌)(Hom ‘𝑇)(𝑂‘𝑍)))
83	81, 82	syl 17	. . . 4 ⊢ (𝜑 → (𝑌𝑃𝑍):(𝑌𝐻𝑍)⟶((𝑂‘𝑌)(Hom ‘𝑇)(𝑂‘𝑍)))
84	83, 29	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → ((𝑌𝑃𝑍)‘𝑁) ∈ ((𝑂‘𝑌)(Hom ‘𝑇)(𝑂‘𝑍)))
85	63, 64, 65, 66, 67, 68, 74, 75, 76, 80, 84	xpcco 18120	. 2 ⊢ (𝜑 → (((𝑌𝑃𝑍)‘𝑁)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝑀)) = ⟨((1^st ‘((𝑌𝑃𝑍)‘𝑁))(⟨(1^st ‘(𝑂‘𝑋)), (1^st ‘(𝑂‘𝑌))⟩(comp‘𝐷)(1^st ‘(𝑂‘𝑍)))(1^st ‘((𝑋𝑃𝑌)‘𝑀))), ((2^nd ‘((𝑌𝑃𝑍)‘𝑁))(⟨(2^nd ‘(𝑂‘𝑋)), (2^nd ‘(𝑂‘𝑌))⟩(comp‘𝐶)(2^nd ‘(𝑂‘𝑍)))(2^nd ‘((𝑋𝑃𝑌)‘𝑀)))⟩)
86		swapfcoa.os	. . . . 5 ⊢ · = (comp‘𝑆)
87	2, 3, 27, 67, 66, 86, 4, 11, 19, 36, 29	xpcco 18120	. . . 4 ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀) = ⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩)
88	87	fveq2d 6844	. . 3 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = ((𝑋𝑃𝑍)‘⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩))
89	2, 69, 70	xpccat 18127	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Cat)
90	3, 27, 86, 89, 4, 11, 19, 36, 29	catcocl 17622	. . . . . 6 ⊢ (𝜑 → (𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀) ∈ (𝑋𝐻𝑍))
91	87, 90	eqeltrrd 2829	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩ ∈ (𝑋𝐻𝑍))
92	1, 2, 3, 4, 19, 28, 91	swapf2a 49233	. . . 4 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩) = ⟨(2^nd ‘⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩), (1^st ‘⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩)⟩)
93		ovex 7402	. . . . . 6 ⊢ ((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)) ∈ V
94		ovex 7402	. . . . . 6 ⊢ ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀)) ∈ V
95	93, 94	op2nd 7956	. . . . 5 ⊢ (2^nd ‘⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩) = ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))
96	93, 94	op1st 7955	. . . . 5 ⊢ (1^st ‘⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩) = ((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀))
97	95, 96	opeq12i 4838	. . . 4 ⊢ ⟨(2^nd ‘⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩), (1^st ‘⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩)⟩ = ⟨((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀)), ((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀))⟩
98	92, 97	eqtrdi 2780	. . 3 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘⟨((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀)), ((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀))⟩) = ⟨((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀)), ((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀))⟩)
99	88, 98	eqtrd 2764	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = ⟨((2^nd ‘𝑁)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝑀)), ((1^st ‘𝑁)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩(comp‘𝐶)(1^st ‘𝑍))(1^st ‘𝑀))⟩)
100	62, 85, 99	3eqtr4rd 2775	1 ⊢ (𝜑 → ((𝑋𝑃𝑍)‘(𝑁(⟨𝑋, 𝑌⟩ · 𝑍)𝑀)) = (((𝑌𝑃𝑍)‘𝑁)(⟨(𝑂‘𝑋), (𝑂‘𝑌)⟩ ∙ (𝑂‘𝑍))((𝑋𝑃𝑌)‘𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ⟨cop 4591 ⟶wf 6495 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 Basecbs 17155 Hom chom 17207 compcco 17208 Catccat 17601 ×_c cxpc 18105 swap_F cswapf 49221

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-tp 4590 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-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 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-uz 12770 df-fz 13445 df-struct 17093 df-slot 17128 df-ndx 17140 df-base 17156 df-hom 17220 df-cco 17221 df-cat 17605 df-cid 17606 df-xpc 18109 df-swapf 49222

This theorem is referenced by: swapffunc 49244

W3C validator