swapf2f1oa - Mathbox for Zhi Wang

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Theorem swapf2f1oa 49282

Description: The morphism part of the swap functor is a bijection between hom-sets. (Contributed by Zhi Wang, 9-Oct-2025.)

**Hypotheses**
Ref	Expression
swapf1f1o.o	⊢ (𝜑 → (𝐶 swap_F 𝐷) = ⟨𝑂, 𝑃⟩)
swapf1f1o.s	⊢ 𝑆 = (𝐶 ×_c 𝐷)
swapf1f1o.t	⊢ 𝑇 = (𝐷 ×_c 𝐶)
swapf2f1o.h	⊢ 𝐻 = (Hom ‘𝑆)
swapf2f1o.j	⊢ 𝐽 = (Hom ‘𝑇)
swapf2f1oa.b	⊢ 𝐵 = (Base‘𝑆)
swapf2f1oa.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
swapf2f1oa.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
swapf2f1oa	⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌)))

**Proof of Theorem swapf2f1oa**
Step	Hyp	Ref	Expression
1		swapf1f1o.o	. . 3 ⊢ (𝜑 → (𝐶 swap_F 𝐷) = ⟨𝑂, 𝑃⟩)
2		swapf1f1o.s	. . 3 ⊢ 𝑆 = (𝐶 ×_c 𝐷)
3		swapf1f1o.t	. . 3 ⊢ 𝑇 = (𝐷 ×_c 𝐶)
4		swapf2f1o.h	. . 3 ⊢ 𝐻 = (Hom ‘𝑆)
5		swapf2f1o.j	. . 3 ⊢ 𝐽 = (Hom ‘𝑇)
6		swapf2f1oa.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		swapf2f1oa.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
8		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	2, 8, 9	xpcbas 18103	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑆)
11	7, 10	eqtr4i 2755	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
12	6, 11	eleqtrdi 2838	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)))
13		xp1st 7963	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1^st ‘𝑋) ∈ (Base‘𝐶))
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → (1^st ‘𝑋) ∈ (Base‘𝐶))
15		xp2nd 7964	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2^nd ‘𝑋) ∈ (Base‘𝐷))
16	12, 15	syl 17	. . 3 ⊢ (𝜑 → (2^nd ‘𝑋) ∈ (Base‘𝐷))
17		swapf2f1oa.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18	17, 11	eleqtrdi 2838	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐶) × (Base‘𝐷)))
19		xp1st 7963	. . . 4 ⊢ (𝑌 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1^st ‘𝑌) ∈ (Base‘𝐶))
20	18, 19	syl 17	. . 3 ⊢ (𝜑 → (1^st ‘𝑌) ∈ (Base‘𝐶))
21		xp2nd 7964	. . . 4 ⊢ (𝑌 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2^nd ‘𝑌) ∈ (Base‘𝐷))
22	18, 21	syl 17	. . 3 ⊢ (𝜑 → (2^nd ‘𝑌) ∈ (Base‘𝐷))
23	1, 2, 3, 4, 5, 14, 16, 20, 22	swapf2f1o 49281	. 2 ⊢ (𝜑 → (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝑃⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩):(⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝐻⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)–1-1-onto→(⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩𝐽⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩))
24		1st2nd2 7970	. . . . 5 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
25	12, 24	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩)
26		1st2nd2 7970	. . . . 5 ⊢ (𝑌 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
27	18, 26	syl 17	. . . 4 ⊢ (𝜑 → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
28	25, 27	oveq12d 7371	. . 3 ⊢ (𝜑 → (𝑋𝑃𝑌) = (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝑃⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩))
29	25, 27	oveq12d 7371	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝐻⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩))
30	1, 2, 7, 6	swapf1a 49274	. . . 4 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩)
31	1, 2, 7, 17	swapf1a 49274	. . . 4 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩)
32	30, 31	oveq12d 7371	. . 3 ⊢ (𝜑 → ((𝑂‘𝑋)𝐽(𝑂‘𝑌)) = (⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩𝐽⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩))
33	28, 29, 32	f1oeq123d 6762	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌)) ↔ (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝑃⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩):(⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝐻⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)–1-1-onto→(⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩𝐽⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩)))
34	23, 33	mpbird 257	1 ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)–1-1-onto→((𝑂‘𝑋)𝐽(𝑂‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ⟨cop 4585 × cxp 5621 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7353 1^st c1st 7929 2^nd c2nd 7930 Basecbs 17139 Hom chom 17191 ×_c cxpc 18093 swap_F cswapf 49264

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 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12611 df-uz 12755 df-fz 13430 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17140 df-hom 17204 df-cco 17205 df-xpc 18097 df-swapf 49265

This theorem is referenced by: swapfcoa 49286 swapffunc 49287 swapfffth 49288

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