Theorem cofuswapf1 49279

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 7352	. . . 4 ⊢ (𝑋(1st ‘𝐺)𝑌) = ((1st ‘𝐺)‘⟨𝑋, 𝑌⟩)
2		cofuswapf1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷)))
3	2	fveq2d 6826	. . . . 5 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘(𝐹 ∘func (𝐶 swapF 𝐷))))
4	3	fveq1d 6824	. . . 4 ⊢ (𝜑 → ((1st ‘𝐺)‘⟨𝑋, 𝑌⟩) = ((1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))‘⟨𝑋, 𝑌⟩))
5	1, 4	eqtrid 2776	. . 3 ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = ((1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))‘⟨𝑋, 𝑌⟩))
6		eqid 2729	. . . . 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 18084	. . . 4 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
10		cofuswapf1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		cofuswapf1.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
12		eqid 2729	. . . . 5 ⊢ (𝐷 ×c 𝐶) = (𝐷 ×c 𝐶)
13	10, 11, 6, 12	swapffunca 49269	. . . 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 5658	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
18	9, 13, 14, 17	cofu1 17791	. . 3 ⊢ (𝜑 → ((1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st ‘𝐹)‘((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)))
19		df-ov 7352	. . . . 5 ⊢ (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)
20	10, 11	swapfelvv 49248	. . . . . . 7 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))
21		1st2nd2 7963	. . . . . . 7 ⊢ ((𝐶 swapF 𝐷) ∈ (V × V) → (𝐶 swapF 𝐷) = ⟨(1st ‘(𝐶 swapF 𝐷)), (2nd ‘(𝐶 swapF 𝐷))⟩)
22	20, 21	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(1st ‘(𝐶 swapF 𝐷)), (2nd ‘(𝐶 swapF 𝐷))⟩)
23	15, 7	eleqtrdi 2838	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
24	16, 8	eleqtrdi 2838	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
25	22, 23, 24	swapf1 49257	. . . . 5 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ⟨𝑌, 𝑋⟩)
26	19, 25	eqtr3id 2778	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩) = ⟨𝑌, 𝑋⟩)
27	26	fveq2d 6826	. . 3 ⊢ (𝜑 → ((1st ‘𝐹)‘((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st ‘𝐹)‘⟨𝑌, 𝑋⟩))
28	5, 18, 27	3eqtrd 2768	. 2 ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = ((1st ‘𝐹)‘⟨𝑌, 𝑋⟩))
29		df-ov 7352	. 2 ⊢ (𝑌(1st ‘𝐹)𝑋) = ((1st ‘𝐹)‘⟨𝑌, 𝑋⟩)
30	28, 29	eqtr4di 2782	1 ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = (𝑌(1st ‘𝐹)𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ⟨cop 4583   × cxp 5617  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  Basecbs 17120  Catccat 17570   Func cfunc 17761   ∘_func ccofu 17763   ×_c cxpc 18074   swap_F cswapf 49244

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-hom 17185  df-cco 17186  df-cat 17574  df-cid 17575  df-func 17765  df-cofu 17767  df-xpc 18078  df-swapf 49245

This theorem is referenced by:  tposcurf11  49282