Theorem swapfida 50115

Description: Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfid 50114. (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 2765	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	3, 8, 9	xpcbas 18256	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑆)
11	7, 10	eqtr4i 2791	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
12	6, 11	eleqtrdi 2875	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)))
13		xp1st 8024	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st ‘𝑋) ∈ (Base‘𝐶))
14	12, 13	syl 18	. . 3 ⊢ (𝜑 → (1st ‘𝑋) ∈ (Base‘𝐶))
15		xp2nd 8025	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2nd ‘𝑋) ∈ (Base‘𝐷))
16	12, 15	syl 18	. . 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 50114	. 2 ⊢ (𝜑 → ((⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)‘( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)) = (𝐼‘(𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
20		1st2nd2 8031	. . . . 5 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
21	12, 20	syl 18	. . . 4 ⊢ (𝜑 → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
22	21, 21	oveq12d 7437	. . 3 ⊢ (𝜑 → (𝑋𝑃𝑋) = (⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
23	21	fveq2d 6889	. . 3 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
24	22, 23	fveq12d 6892	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = ((⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)‘( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
25	21	fveq2d 6889	. . 3 ⊢ (𝜑 → (𝑂‘𝑋) = (𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
26	25	fveq2d 6889	. 2 ⊢ (𝜑 → (𝐼‘(𝑂‘𝑋)) = (𝐼‘(𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
27	19, 24, 26	3eqtr4d 2810	1 ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝑂‘𝑋)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ⟨cop 4597   × cxp 5661  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  2^nd c2nd 7991  Basecbs 17291  Catccat 17742  Idccid 17743   ×_c cxpc 18246   swap_F cswapf 50094

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-z 12607  df-dec 12728  df-uz 12879  df-fz 13552  df-struct 17229  df-slot 17264  df-ndx 17276  df-base 17292  df-hom 17356  df-cco 17357  df-cat 17746  df-cid 17747  df-xpc 18250  df-swapf 50095

This theorem is used by:  swapffunc  50117