Theorem swapfida 48959

Description: Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfid 48958. (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⟩)
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 → (𝐶swapF𝐷) = ⟨𝑂, 𝑃⟩)
6		swapfida.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		swapfida.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
8		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	3, 8, 9	xpcbas 18219	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑆)
11	7, 10	eqtr4i 2767	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
12	6, 11	eleqtrdi 2850	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)))
13		xp1st 8042	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st ‘𝑋) ∈ (Base‘𝐶))
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → (1st ‘𝑋) ∈ (Base‘𝐶))
15		xp2nd 8043	. . . 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 48958	. 2 ⊢ (𝜑 → ((⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)‘( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)) = (𝐼‘(𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
20		1st2nd2 8049	. . . . 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 7447	. . 3 ⊢ (𝜑 → (𝑋𝑃𝑋) = (⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
23	21	fveq2d 6908	. . 3 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
24	22, 23	fveq12d 6911	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = ((⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)‘( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
25	21	fveq2d 6908	. . 3 ⊢ (𝜑 → (𝑂‘𝑋) = (𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
26	25	fveq2d 6908	. 2 ⊢ (𝜑 → (𝐼‘(𝑂‘𝑋)) = (𝐼‘(𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
27	19, 24, 26	3eqtr4d 2786	1 ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝑂‘𝑋)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ⟨cop 4630   × cxp 5681  ‘cfv 6559  (class class class)co 7429  1^st c1st 8008  2^nd c2nd 8009  Basecbs 17243  Catccat 17703  Idccid 17704   ×_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-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-xpc 18213  df-swapf 48939

This theorem is referenced by:  swapffunc  48961