Theorem swapffunc 48961

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 → (𝐶swapF𝐷) = ⟨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 2736	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2736	. 2 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2736	. 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
4		eqid 2736	. 2 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
5		eqid 2736	. 2 ⊢ (Id‘𝑆) = (Id‘𝑆)
6		eqid 2736	. 2 ⊢ (Id‘𝑇) = (Id‘𝑇)
7		eqid 2736	. 2 ⊢ (comp‘𝑆) = (comp‘𝑆)
8		eqid 2736	. 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 18231	. 2 ⊢ (𝜑 → 𝑆 ∈ Cat)
13		swapfid.t	. . 3 ⊢ 𝑇 = (𝐷 ×c 𝐶)
14	13, 11, 10	xpccat 18231	. 2 ⊢ (𝜑 → 𝑇 ∈ Cat)
15		swapfid.o	. . . 4 ⊢ (𝜑 → (𝐶swapF𝐷) = ⟨𝑂, 𝑃⟩)
16	15, 9, 13, 10, 11, 1, 2	swapf1f1o 48954	. . 3 ⊢ (𝜑 → 𝑂:(Base‘𝑆)–1-1-onto→(Base‘𝑇))
17		f1of 6846	. . 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 48947	. 2 ⊢ (𝜑 → 𝑃 Fn ((Base‘𝑆) × (Base‘𝑆)))
20	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐶swapF𝐷) = ⟨𝑂, 𝑃⟩)
21		simprl 771	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
22		simprr 773	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
23	20, 9, 13, 3, 4, 1, 21, 22	swapf2f1oa 48956	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥𝑃𝑦):(𝑥(Hom ‘𝑆)𝑦)–1-1-onto→((𝑂‘𝑥)(Hom ‘𝑇)(𝑂‘𝑦)))
24		f1of 6846	. . 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)) → (𝐶swapF𝐷) = ⟨𝑂, 𝑃⟩)
29		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
30	26, 27, 9, 13, 28, 1, 29, 5, 6	swapfida 48959	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥𝑃𝑥)‘((Id‘𝑆)‘𝑥)) = ((Id‘𝑇)‘(𝑂‘𝑥)))
31	10	3ad2ant1 1134	. . 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 1134	. . 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 1134	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) ∧ (𝑚 ∈ (𝑥(Hom ‘𝑆)𝑦) ∧ 𝑛 ∈ (𝑦(Hom ‘𝑆)𝑧))) → (𝐶swapF𝐷) = ⟨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 48960	. 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 17906	1 ⊢ (𝜑 → 𝑂(𝑆 Func 𝑇)𝑃)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ⟨cop 4630   class class class wbr 5141  ⟶wf 6555  –1-1-onto→wf1o 6558  ‘cfv 6559  (class class class)co 7429  Basecbs 17243  Hom chom 17304  compcco 17305  Catccat 17703  Idccid 17704   Func cfunc 17895   ×_c cxpc 18209  swap_Fcswapf 48938

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751  ax-cnex 11207  ax-resscn 11208  ax-1cn 11209  ax-icn 11210  ax-addcl 11211  ax-addrcl 11212  ax-mulcl 11213  ax-mulrcl 11214  ax-mulcom 11215  ax-addass 11216  ax-mulass 11217  ax-distr 11218  ax-i2m1 11219  ax-1ne0 11220  ax-1rid 11221  ax-rnegex 11222  ax-rrecex 11223  ax-cnre 11224  ax-pre-lttri 11225  ax-pre-lttrn 11226  ax-pre-ltadd 11227  ax-pre-mulgt0 11228

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-om 7884  df-1st 8010  df-2nd 8011  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-1o 8502  df-er 8741  df-map 8864  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-pnf 11293  df-mnf 11294  df-xr 11295  df-ltxr 11296  df-le 11297  df-sub 11490  df-neg 11491  df-nn 12263  df-2 12325  df-3 12326  df-4 12327  df-5 12328  df-6 12329  df-7 12330  df-8 12331  df-9 12332  df-n0 12523  df-z 12610  df-dec 12730  df-uz 12875  df-fz 13544  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17244  df-hom 17317  df-cco 17318  df-cat 17707  df-cid 17708  df-func 17899  df-xpc 18213  df-swapf 48939

This theorem is referenced by:  swapfffth  48962  swapffunca  48963