Theorem swapffunc 49443

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 2733	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2733	. 2 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2733	. 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
4		eqid 2733	. 2 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
5		eqid 2733	. 2 ⊢ (Id‘𝑆) = (Id‘𝑆)
6		eqid 2733	. 2 ⊢ (Id‘𝑇) = (Id‘𝑇)
7		eqid 2733	. 2 ⊢ (comp‘𝑆) = (comp‘𝑆)
8		eqid 2733	. 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 18104	. 2 ⊢ (𝜑 → 𝑆 ∈ Cat)
13		swapfid.t	. . 3 ⊢ 𝑇 = (𝐷 ×c 𝐶)
14	13, 11, 10	xpccat 18104	. 2 ⊢ (𝜑 → 𝑇 ∈ Cat)
15		swapfid.o	. . . 4 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
16	15, 9, 13, 10, 11, 1, 2	swapf1f1o 49436	. . 3 ⊢ (𝜑 → 𝑂:(Base‘𝑆)–1-1-onto→(Base‘𝑇))
17		f1of 6771	. . 3 ⊢ (𝑂:(Base‘𝑆)–1-1-onto→(Base‘𝑇) → 𝑂:(Base‘𝑆)⟶(Base‘𝑇))
18	16, 17	syl 17	. 2 ⊢ (𝜑 → 𝑂:(Base‘𝑆)⟶(Base‘𝑇))
19	10, 11, 9, 1, 15	swapf2fn 49429	. 2 ⊢ (𝜑 → 𝑃 Fn ((Base‘𝑆) × (Base‘𝑆)))
20	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
21		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
22		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
23	20, 9, 13, 3, 4, 1, 21, 22	swapf2f1oa 49438	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥𝑃𝑦):(𝑥(Hom ‘𝑆)𝑦)–1-1-onto→((𝑂‘𝑥)(Hom ‘𝑇)(𝑂‘𝑦)))
24		f1of 6771	. . 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 17	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥𝑃𝑦):(𝑥(Hom ‘𝑆)𝑦)⟶((𝑂‘𝑥)(Hom ‘𝑇)(𝑂‘𝑦)))
26	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐶 ∈ Cat)
27	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐷 ∈ Cat)
28	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
29		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
30	26, 27, 9, 13, 28, 1, 29, 5, 6	swapfida 49441	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥𝑃𝑥)‘((Id‘𝑆)‘𝑥)) = ((Id‘𝑇)‘(𝑂‘𝑥)))
31	10	3ad2ant1 1133	. . 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 1133	. . 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 1133	. . 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 1207	. . 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 1208	. . 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 1209	. . 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 1202	. . 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 1203	. . 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 49442	. 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 17780	1 ⊢ (𝜑 → 𝑂(𝑆 Func 𝑇)𝑃)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ⟨cop 4583   class class class wbr 5095  ⟶wf 6485  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7355  Basecbs 17127  Hom chom 17179  compcco 17180  Catccat 17578  Idccid 17579   Func cfunc 17769   ×_c cxpc 18082   swap_F cswapf 49420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-map 8761  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-3 12200  df-4 12201  df-5 12202  df-6 12203  df-7 12204  df-8 12205  df-9 12206  df-n0 12393  df-z 12480  df-dec 12599  df-uz 12743  df-fz 13415  df-struct 17065  df-slot 17100  df-ndx 17112  df-base 17128  df-hom 17192  df-cco 17193  df-cat 17582  df-cid 17583  df-func 17773  df-xpc 18086  df-swapf 49421

This theorem is referenced by:  swapfffth  49444  swapffunca  49445