Theorem swapfida 49441

Description: Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfid 49440. (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⟩)
swapfida.b	⊢ 𝐵 = (Base‘𝑆)
swapfida.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
swapfida.1	⊢ 1 = (Id‘𝑆)
swapfida.i	⊢ 𝐼 = (Id‘𝑇)

Assertion

Ref	Expression
swapfida	⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝑂‘𝑋)))

Proof of Theorem swapfida

Step	Hyp	Ref	Expression
1		swapfid.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
2		swapfid.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
3		swapfid.s	. . 3 ⊢ 𝑆 = (𝐶 ×c 𝐷)
4		swapfid.t	. . 3 ⊢ 𝑇 = (𝐷 ×c 𝐶)
5		swapfid.o	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
6		swapfida.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		swapfida.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
8		eqid 2733	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2733	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	3, 8, 9	xpcbas 18092	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑆)
11	7, 10	eqtr4i 2759	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
12	6, 11	eleqtrdi 2843	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)))
13		xp1st 7962	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st ‘𝑋) ∈ (Base‘𝐶))
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → (1st ‘𝑋) ∈ (Base‘𝐶))
15		xp2nd 7963	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2nd ‘𝑋) ∈ (Base‘𝐷))
16	12, 15	syl 17	. . 3 ⊢ (𝜑 → (2nd ‘𝑋) ∈ (Base‘𝐷))
17		swapfida.1	. . 3 ⊢ 1 = (Id‘𝑆)
18		swapfida.i	. . 3 ⊢ 𝐼 = (Id‘𝑇)
19	1, 2, 3, 4, 5, 14, 16, 17, 18	swapfid 49440	. 2 ⊢ (𝜑 → ((⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)‘( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)) = (𝐼‘(𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
20		1st2nd2 7969	. . . . 5 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
21	12, 20	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
22	21, 21	oveq12d 7373	. . 3 ⊢ (𝜑 → (𝑋𝑃𝑋) = (⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
23	21	fveq2d 6835	. . 3 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
24	22, 23	fveq12d 6838	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = ((⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)‘( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
25	21	fveq2d 6835	. . 3 ⊢ (𝜑 → (𝑂‘𝑋) = (𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
26	25	fveq2d 6835	. 2 ⊢ (𝜑 → (𝐼‘(𝑂‘𝑋)) = (𝐼‘(𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
27	19, 24, 26	3eqtr4d 2778	1 ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝑂‘𝑋)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4583   × cxp 5619  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929  Basecbs 17127  Catccat 17578  Idccid 17579   ×_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-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-xpc 18086  df-swapf 49421

This theorem is referenced by:  swapffunc  49443