Theorem swapffunc 49841

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 2752	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2752	. 2 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		eqid 2752	. 2 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
4		eqid 2752	. 2 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
5		eqid 2752	. 2 ⊢ (Id‘𝑆) = (Id‘𝑆)
6		eqid 2752	. 2 ⊢ (Id‘𝑇) = (Id‘𝑇)
7		eqid 2752	. 2 ⊢ (comp‘𝑆) = (comp‘𝑆)
8		eqid 2752	. 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 18194	. 2 ⊢ (𝜑 → 𝑆 ∈ Cat)
13		swapfid.t	. . 3 ⊢ 𝑇 = (𝐷 ×c 𝐶)
14	13, 11, 10	xpccat 18194	. 2 ⊢ (𝜑 → 𝑇 ∈ Cat)
15		swapfid.o	. . . 4 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
16	15, 9, 13, 10, 11, 1, 2	swapf1f1o 49834	. . 3 ⊢ (𝜑 → 𝑂:(Base‘𝑆)–1-1-onto→(Base‘𝑇))
17		f1of 6791	. . 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 49827	. 2 ⊢ (𝜑 → 𝑃 Fn ((Base‘𝑆) × (Base‘𝑆)))
20	15	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
21		simprl 778	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
22		simprr 780	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
23	20, 9, 13, 3, 4, 1, 21, 22	swapf2f1oa 49836	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥𝑃𝑦):(𝑥(Hom ‘𝑆)𝑦)–1-1-onto→((𝑂‘𝑥)(Hom ‘𝑇)(𝑂‘𝑦)))
24		f1of 6791	. . 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 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐶 ∈ Cat)
27	11	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐷 ∈ Cat)
28	15	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
29		simpr 487	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
30	26, 27, 9, 13, 28, 1, 29, 5, 6	swapfida 49839	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝑥𝑃𝑥)‘((Id‘𝑆)‘𝑥)) = ((Id‘𝑇)‘(𝑂‘𝑥)))
31	10	3ad2ant1 1142	. . 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 1142	. . 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 1142	. . 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 1216	. . 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 1217	. . 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 1218	. . 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 1211	. . 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 1212	. . 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 49840	. 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 17870	1 ⊢ (𝜑 → 𝑂(𝑆 Func 𝑇)𝑃)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132  ⟨cop 4578   class class class wbr 5090  ⟶wf 6502  –1-1-onto→wf1o 6505  ‘cfv 6506  (class class class)co 7381  Basecbs 17217  Hom chom 17269  compcco 17270  Catccat 17668  Idccid 17669   Func cfunc 17859   ×_c cxpc 18172   swap_F cswapf 49818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-er 8662  df-map 8794  df-ixp 8865  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-nn 12197  df-2 12266  df-3 12267  df-4 12268  df-5 12269  df-6 12270  df-7 12271  df-8 12272  df-9 12273  df-n0 12468  df-z 12555  df-dec 12675  df-uz 12826  df-fz 13499  df-struct 17155  df-slot 17190  df-ndx 17202  df-base 17218  df-hom 17282  df-cco 17283  df-cat 17672  df-cid 17673  df-func 17863  df-xpc 18176  df-swapf 49819

This theorem is referenced by:  swapfffth  49842  swapffunca  49843