Theorem cofuswapf1 49539

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 7361	. . . 4 ⊢ (𝑋(1st ‘𝐺)𝑌) = ((1st ‘𝐺)‘⟨𝑋, 𝑌⟩)
2		cofuswapf1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝐹 ∘func (𝐶 swapF 𝐷)))
3	2	fveq2d 6838	. . . . 5 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘(𝐹 ∘func (𝐶 swapF 𝐷))))
4	3	fveq1d 6836	. . . 4 ⊢ (𝜑 → ((1st ‘𝐺)‘⟨𝑋, 𝑌⟩) = ((1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))‘⟨𝑋, 𝑌⟩))
5	1, 4	eqtrid 2783	. . 3 ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = ((1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))‘⟨𝑋, 𝑌⟩))
6		eqid 2736	. . . . 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 18101	. . . 4 ⊢ (𝐴 × 𝐵) = (Base‘(𝐶 ×c 𝐷))
10		cofuswapf1.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
11		cofuswapf1.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
12		eqid 2736	. . . . 5 ⊢ (𝐷 ×c 𝐶) = (𝐷 ×c 𝐶)
13	10, 11, 6, 12	swapffunca 49529	. . . 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 5663	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐴 × 𝐵))
18	9, 13, 14, 17	cofu1 17808	. . 3 ⊢ (𝜑 → ((1st ‘(𝐹 ∘func (𝐶 swapF 𝐷)))‘⟨𝑋, 𝑌⟩) = ((1st ‘𝐹)‘((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)))
19		df-ov 7361	. . . . 5 ⊢ (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)
20	10, 11	swapfelvv 49508	. . . . . . 7 ⊢ (𝜑 → (𝐶 swapF 𝐷) ∈ (V × V))
21		1st2nd2 7972	. . . . . . 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 2846	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
24	16, 8	eleqtrdi 2846	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
25	22, 23, 24	swapf1 49517	. . . . 5 ⊢ (𝜑 → (𝑋(1st ‘(𝐶 swapF 𝐷))𝑌) = ⟨𝑌, 𝑋⟩)
26	19, 25	eqtr3id 2785	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩) = ⟨𝑌, 𝑋⟩)
27	26	fveq2d 6838	. . 3 ⊢ (𝜑 → ((1st ‘𝐹)‘((1st ‘(𝐶 swapF 𝐷))‘⟨𝑋, 𝑌⟩)) = ((1st ‘𝐹)‘⟨𝑌, 𝑋⟩))
28	5, 18, 27	3eqtrd 2775	. 2 ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = ((1st ‘𝐹)‘⟨𝑌, 𝑋⟩))
29		df-ov 7361	. 2 ⊢ (𝑌(1st ‘𝐹)𝑋) = ((1st ‘𝐹)‘⟨𝑌, 𝑋⟩)
30	28, 29	eqtr4di 2789	1 ⊢ (𝜑 → (𝑋(1st ‘𝐺)𝑌) = (𝑌(1st ‘𝐹)𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ⟨cop 4586   × cxp 5622  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17136  Catccat 17587   Func cfunc 17778   ∘_func ccofu 17780   ×_c cxpc 18091   swap_F cswapf 49504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-hom 17201  df-cco 17202  df-cat 17591  df-cid 17592  df-func 17782  df-cofu 17784  df-xpc 18095  df-swapf 49505

This theorem is referenced by:  tposcurf11  49542