swapf2f1oa - Mathbox for Zhi Wang

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Theorem swapf2f1oa 49752

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 2736	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	2, 8, 9	xpcbas 18144	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑆)
11	7, 10	eqtr4i 2762	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
12	6, 11	eleqtrdi 2846	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)))
13		xp1st 7974	. . . 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 7975	. . . 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 2846	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐶) × (Base‘𝐷)))
19		xp1st 7974	. . . 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 7975	. . . 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 49751	. 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 7981	. . . . 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 7981	. . . . 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 7385	. . 3 ⊢ (𝜑 → (𝑋𝑃𝑌) = (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝑃⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩))
29	25, 27	oveq12d 7385	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝐻⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩))
30	1, 2, 7, 6	swapf1a 49744	. . . 4 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩)
31	1, 2, 7, 17	swapf1a 49744	. . . 4 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩)
32	30, 31	oveq12d 7385	. . 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 6774	. 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 1542 ∈ wcel 2114 ⟨cop 4573 × cxp 5629 –1-1-onto→wf1o 6497 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 Basecbs 17179 Hom chom 17231 ×_c cxpc 18134 swap_F cswapf 49734

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 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 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-hom 17244 df-cco 17245 df-xpc 18138 df-swapf 49735

This theorem is referenced by: swapfcoa 49756 swapffunc 49757 swapfffth 49758

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