swapf2f1oa - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > swapf2f1oa		Structured version Visualization version GIF version

Theorem swapf2f1oa 49239

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 18115	. . . . . 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 7979	. . . 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 7980	. . . 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 7979	. . . 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 7980	. . . 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 49238	. 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 7986	. . . . 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 7986	. . . . 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 7387	. . 3 ⊢ (𝜑 → (𝑋𝑃𝑌) = (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝑃⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩))
29	25, 27	oveq12d 7387	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝐻⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩))
30	1, 2, 7, 6	swapf1a 49231	. . . 4 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩)
31	1, 2, 7, 17	swapf1a 49231	. . . 4 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩)
32	30, 31	oveq12d 7387	. . 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 6776	. 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 4591 × cxp 5629 –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 ×_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-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-xpc 18109 df-swapf 49222

This theorem is referenced by: swapfcoa 49243 swapffunc 49244 swapfffth 49245

W3C validator