Theorem cofuswapf1 50129

Description: The object part of a bifunctor pre-composed with a swap functor. (Contributed by Zhi Wang, 9-Oct-2025.)

Hypotheses

Ref	Expression
cofuswapf1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
cofuswapf1.d	⊢ (𝜑 → 𝐷 ∈ Cat)
cofuswapf1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
cofuswapf1.g	⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷)))
cofuswapf1.a	⊢ 𝐴 = (Base‘𝐶)
cofuswapf1.b	⊢ 𝐵 = (Base‘𝐷)
cofuswapf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
cofuswapf1.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
cofuswapf1	⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = (𝑌(1st ‘𝐹)𝑋))

Proof of Theorem cofuswapf1

Step	Hyp	Ref	Expression
1		df-ov 7422	. . . 4 ⊢ (𝑋(1st ‘𝐺)𝑌) = ((1st ‘𝐺)‘⟨𝑋, 𝑌⟩)
2		cofuswapf1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷)))
3	2	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘(𝐹 ∘func (𝐶 swapF 𝐷))))
4	3	fveq1d 6887	. . . 4 ⊢ (𝜑 → ((1st ‘𝐺)‘⟨𝑋, 𝑌⟩) = ((1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))‘⟨𝑋, 𝑌⟩))
5	1, 4	eqtrid 2812	. . 3 ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = ((1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))‘⟨𝑋, 𝑌⟩))
6		eqid 2765	. . . . 5 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷)
7		cofuswapf1.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
8		cofuswapf1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
9	6, 7, 8	xpcbas 18256	. . . 4 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
10		cofuswapf1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		cofuswapf1.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
12		eqid 2765	. . . . 5 ⊢ (𝐷 ×c 𝐶) = (𝐷 ×c 𝐶)
13	10, 11, 6, 12	swapffunca 50119	. . . 4 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ ((𝐶 ×c 𝐷) Func (𝐷 ×c 𝐶)))
14		cofuswapf1.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
15		cofuswapf1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
16		cofuswapf1.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
17	15, 16	opelxpd 5702	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
18	9, 13, 14, 17	cofu1 17963	. . 3 ⊢ (𝜑 → ((1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st ‘𝐹)‘((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)))
19		df-ov 7422	. . . . 5 ⊢ (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)
20	10, 11	swapfelvv 50098	. . . . . . 7 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))
21		1st2nd2 8031	. . . . . . 7 ⊢ ((𝐶 swapF 𝐷) ∈ (V × V) → (𝐶 swapF 𝐷) = ⟨(1st ‘(𝐶 swapF 𝐷)), (2nd ‘(𝐶 swapF 𝐷))⟩)
22	20, 21	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(1st ‘(𝐶 swapF 𝐷)), (2nd ‘(𝐶 swapF 𝐷))⟩)
23	15, 7	eleqtrdi 2875	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
24	16, 8	eleqtrdi 2875	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
25	22, 23, 24	swapf1 50107	. . . . 5 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ⟨𝑌, 𝑋⟩)
26	19, 25	eqtr3id 2814	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩) = ⟨𝑌, 𝑋⟩)
27	26	fveq2d 6889	. . 3 ⊢ (𝜑 → ((1st ‘𝐹)‘((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st ‘𝐹)‘⟨𝑌, 𝑋⟩))
28	5, 18, 27	3eqtrd 2804	. 2 ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = ((1st ‘𝐹)‘⟨𝑌, 𝑋⟩))
29		df-ov 7422	. 2 ⊢ (𝑌(1st ‘𝐹)𝑋) = ((1st ‘𝐹)‘⟨𝑌, 𝑋⟩)
30	28, 29	eqtr4di 2818	1 ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = (𝑌(1st ‘𝐹)𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597   × cxp 5661  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  2^nd c2nd 7991  Basecbs 17291  Catccat 17742   Func cfunc 17933   ∘_func ccofu 17935   ×_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-map 8832  df-ixp 8902  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-func 17937  df-cofu 17939  df-xpc 18250  df-swapf 50095

This theorem is used by:  tposcurf11  50132