Theorem swapfida 49755

Description: Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also swapfid 49754. (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 2736	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	3, 8, 9	xpcbas 18144	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑆)
11	7, 10	eqtr4i 2762	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
12	6, 11	eleqtrdi 2846	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)))
13		xp1st 7974	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st ‘𝑋) ∈ (Base‘𝐶))
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → (1st ‘𝑋) ∈ (Base‘𝐶))
15		xp2nd 7975	. . . 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 49754	. 2 ⊢ (𝜑 → ((⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)‘( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)) = (𝐼‘(𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
20		1st2nd2 7981	. . . . 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 7385	. . 3 ⊢ (𝜑 → (𝑋𝑃𝑋) = (⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
23	21	fveq2d 6844	. . 3 ⊢ (𝜑 → ( 1 ‘𝑋) = ( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
24	22, 23	fveq12d 6847	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = ((⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)‘( 1 ‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
25	21	fveq2d 6844	. . 3 ⊢ (𝜑 → (𝑂‘𝑋) = (𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
26	25	fveq2d 6844	. 2 ⊢ (𝜑 → (𝐼‘(𝑂‘𝑋)) = (𝐼‘(𝑂‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)))
27	19, 24, 26	3eqtr4d 2781	1 ⊢ (𝜑 → ((𝑋𝑃𝑋)‘( 1 ‘𝑋)) = (𝐼‘(𝑂‘𝑋)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4573   × cxp 5629  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  Basecbs 17179  Catccat 17630  Idccid 17631   ×_c cxpc 18134   swap_F cswapf 49734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17634  df-cid 17635  df-xpc 18138  df-swapf 49735

This theorem is referenced by:  swapffunc  49757