Theorem swapffunc 50060

Description: The swap functor is a functor. (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 swapF 𝐷) = ⟨𝑂, 𝑃⟩)

Assertion

Ref	Expression
swapffunc	⊢ (𝜑 → 𝑂(𝑆 Func 𝑇)𝑃)

Proof of Theorem swapffunc

Dummy variables 𝑚 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2763	. 2 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2763	. 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
4		eqid 2763	. 2 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
5		eqid 2763	. 2 ⊢ (Id‘𝑆) = (Id‘𝑆)
6		eqid 2763	. 2 ⊢ (Id‘𝑇) = (Id‘𝑇)
7		eqid 2763	. 2 ⊢ (comp‘𝑆) = (comp‘𝑆)
8		eqid 2763	. 2 ⊢ (comp‘𝑇) = (comp‘𝑇)
9		swapfid.s	. . 3 ⊢ 𝑆 = (𝐶 ×c 𝐷)
10		swapfid.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		swapfid.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
12	9, 10, 11	xpccat 18241	. 2 ⊢ (𝜑 → 𝑆 ∈ Cat)
13		swapfid.t	. . 3 ⊢ 𝑇 = (𝐷 ×c 𝐶)
14	13, 11, 10	xpccat 18241	. 2 ⊢ (𝜑 → 𝑇 ∈ Cat)
15		swapfid.o	. . . 4 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
16	15, 9, 13, 10, 11, 1, 2	swapf1f1o 50053	. . 3 ⊢ (𝜑 → 𝑂:(Base‘𝑆)–1-1-onto→(Base‘𝑇))
17		f1of 6820	. . 3 ⊢ (𝑂:(Base‘𝑆)–1-1-onto→(Base‘𝑇) → 𝑂:(Base‘𝑆)⟶(Base‘𝑇))
18	16, 17	syl 18	. 2 ⊢ (𝜑 → 𝑂:(Base‘𝑆)⟶(Base‘𝑇))
19	10, 11, 9, 1, 15	swapf2fn 50046	. 2 ⊢ (𝜑 → 𝑃 Fn ((Base‘𝑆) × (Base‘𝑆)))
20	15	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
21		simprl 782	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
22		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
23	20, 9, 13, 3, 4, 1, 21, 22	swapf2f1oa 50055	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥𝑃𝑦):(𝑥(Hom ‘𝑆)𝑦)–1-1-onto→((𝑂‘𝑥)(Hom ‘𝑇)(𝑂‘𝑦)))
24		f1of 6820	. . 3 ⊢ ((𝑥𝑃𝑦):(𝑥(Hom ‘𝑆)𝑦)–1-1-onto→((𝑂‘𝑥)(Hom ‘𝑇)(𝑂‘𝑦)) → (𝑥𝑃𝑦):(𝑥(Hom ‘𝑆)𝑦)⟶((𝑂‘𝑥)(Hom ‘𝑇)(𝑂‘𝑦)))
25	23, 24	syl 18	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥𝑃𝑦):(𝑥(Hom ‘𝑆)𝑦)⟶((𝑂‘𝑥)(Hom ‘𝑇)(𝑂‘𝑦)))
26	10	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐶 ∈ Cat)
27	11	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐷 ∈ Cat)
28	15	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
29		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
30	26, 27, 9, 13, 28, 1, 29, 5, 6	swapfida 50058	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥𝑃𝑥)‘((Id‘𝑆)‘𝑥)) = ((Id‘𝑇)‘(𝑂‘𝑥)))
31	10	3ad2ant1 1151	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))) → 𝐶 ∈ Cat)
32	11	3ad2ant1 1151	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))) → 𝐷 ∈ Cat)
33	15	3ad2ant1 1151	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))) → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
34		simp21 1225	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))) → 𝑥 ∈ (Base‘𝑆))
35		simp22 1226	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))) → 𝑦 ∈ (Base‘𝑆))
36		simp23 1227	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))) → 𝑧 ∈ (Base‘𝑆))
37		simp3l 1220	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))) → 𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦))
38		simp3r 1221	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))) → 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))
39	31, 32, 9, 13, 33, 1, 34, 35, 36, 3, 37, 38, 7, 8	swapfcoa 50059	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))) → ((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝑆)𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩(comp‘𝑇)(𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)))
40	1, 2, 3, 4, 5, 6, 7, 8, 12, 14, 18, 19, 25, 30, 39	isfuncd 17917	1 ⊢ (𝜑 → 𝑂(𝑆 Func 𝑇)𝑃)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ⟨cop 4595   class class class wbr 5109  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  Hom chom 17316  compcco 17317  Catccat 17715  Idccid 17716   Func cfunc 17906   ×_c cxpc 18219   swap_F cswapf 50037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  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 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-fz 13531  df-struct 17202  df-slot 17237  df-ndx 17249  df-base 17265  df-hom 17329  df-cco 17330  df-cat 17719  df-cid 17720  df-func 17910  df-xpc 18223  df-swapf 50038

This theorem is referenced by:  swapfffth  50061  swapffunca  50062