swapf2f1oa - Mathbox for Zhi Wang

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Theorem swapf2f1oa 49248

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 2730	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2730	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	2, 8, 9	xpcbas 18145	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑆)
11	7, 10	eqtr4i 2756	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
12	6, 11	eleqtrdi 2839	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)))
13		xp1st 8002	. . . 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 8003	. . . 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 2839	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐶) × (Base‘𝐷)))
19		xp1st 8002	. . . 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 8003	. . . 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 49247	. 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 8009	. . . . 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 8009	. . . . 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 7407	. . 3 ⊢ (𝜑 → (𝑋𝑃𝑌) = (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝑃⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩))
29	25, 27	oveq12d 7407	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩𝐻⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩))
30	1, 2, 7, 6	swapf1a 49240	. . . 4 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2^nd ‘𝑋), (1^st ‘𝑋)⟩)
31	1, 2, 7, 17	swapf1a 49240	. . . 4 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2^nd ‘𝑌), (1^st ‘𝑌)⟩)
32	30, 31	oveq12d 7407	. . 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 6796	. 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 4597 × cxp 5638 –1-1-onto→wf1o 6512 ‘cfv 6513 (class class class)co 7389 1^st c1st 7968 2^nd c2nd 7969 Basecbs 17185 Hom chom 17237 ×_c cxpc 18135 swap_F cswapf 49230

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-uz 12800 df-fz 13475 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17186 df-hom 17250 df-cco 17251 df-xpc 18139 df-swapf 49231

This theorem is referenced by: swapfcoa 49252 swapffunc 49253 swapfffth 49254

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