Theorem swapffunc 49940

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 2769	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2769	. 2 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2769	. 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
4		eqid 2769	. 2 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
5		eqid 2769	. 2 ⊢ (Id‘𝑆) = (Id‘𝑆)
6		eqid 2769	. 2 ⊢ (Id‘𝑇) = (Id‘𝑇)
7		eqid 2769	. 2 ⊢ (comp‘𝑆) = (comp‘𝑆)
8		eqid 2769	. 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 18242	. 2 ⊢ (𝜑 → 𝑆 ∈ Cat)
13		swapfid.t	. . 3 ⊢ 𝑇 = (𝐷 ×c 𝐶)
14	13, 11, 10	xpccat 18242	. 2 ⊢ (𝜑 → 𝑇 ∈ Cat)
15		swapfid.o	. . . 4 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
16	15, 9, 13, 10, 11, 1, 2	swapf1f1o 49933	. . 3 ⊢ (𝜑 → 𝑂:(Base‘𝑆)–1-1-onto→(Base‘𝑇))
17		f1of 6818	. . 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 49926	. 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 49935	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥𝑃𝑦):(𝑥(Hom ‘𝑆)𝑦)–1-1-onto→((𝑂‘𝑥)(Hom ‘𝑇)(𝑂‘𝑦)))
24		f1of 6818	. . 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 49938	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥𝑃𝑥)‘((Id‘𝑆)‘𝑥)) = ((Id‘𝑇)‘(𝑂‘𝑥)))
31	10	3ad2ant1 1149	. . 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 1149	. . 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 1149	. . 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 1223	. . 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 1224	. . 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 1225	. . 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 1218	. . 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 1219	. . 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 49939	. 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 17918	1 ⊢ (𝜑 → 𝑂(𝑆 Func 𝑇)𝑃)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ⟨cop 4597   class class class wbr 5110  ⟶wf 6530  –1-1-onto→wf1o 6533  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  Hom chom 17317  compcco 17318  Catccat 17716  Idccid 17717   Func cfunc 17907   ×_c cxpc 18220   swap_F cswapf 49917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  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 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-z 12588  df-dec 12708  df-uz 12859  df-fz 13532  df-struct 17203  df-slot 17238  df-ndx 17250  df-base 17266  df-hom 17330  df-cco 17331  df-cat 17720  df-cid 17721  df-func 17911  df-xpc 18224  df-swapf 49918

This theorem is referenced by:  swapfffth  49941  swapffunca  49942