Theorem tposcurf12 49283

Description: The partially evaluated transposed curry functor at a morphism. (Contributed by Zhi Wang, 9-Oct-2025.)

Hypotheses

Ref	Expression
tposcurf1.g	⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))
tposcurf1.a	⊢ 𝐴 = (Base‘𝐶)
tposcurf1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
tposcurf1.d	⊢ (𝜑 → 𝐷 ∈ Cat)
tposcurf1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
tposcurf1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
tposcurf1.k	⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋))
tposcurf1.b	⊢ 𝐵 = (Base‘𝐷)
tposcurf11.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
tposcurf12.j	⊢ 𝐽 = (Hom ‘𝐷)
tposcurf12.1	⊢ 1 = (Id‘𝐶)
tposcurf12.y	⊢ (𝜑 → 𝑍 ∈ 𝐵)
tposcurf12.g	⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍))

Assertion

Ref	Expression
tposcurf12	⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (𝐻(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑍, 𝑋⟩)( 1 ‘𝑋)))

Proof of Theorem tposcurf12

Step	Hyp	Ref	Expression
1		tposcurf1.k	. . . . . 6 ⊢ (𝜑 → 𝐾 = ((1st ‘𝐺)‘𝑋))
2		tposcurf1.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))
3	2	fveq2d 6826	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐺) = (1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))))
4	3	fveq1d 6824	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐺)‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋))
5	1, 4	eqtrd 2764	. . . . 5 ⊢ (𝜑 → 𝐾 = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋))
6	5	fveq2d 6826	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋)))
7	6	oveqd 7366	. . 3 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑍) = (𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋))𝑍))
8	7	fveq1d 6824	. 2 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋))𝑍)‘𝐻))
9		eqid 2729	. . 3 ⊢ (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))) = (⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷)))
10		tposcurf1.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
11		tposcurf1.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
12		tposcurf1.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
13		tposcurf1.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐷 ×c 𝐶) Func 𝐸))
14		eqidd 2730	. . . 4 ⊢ (𝜑 → (𝐹 ∘func (𝐶 swapF 𝐷)) = (𝐹 ∘func (𝐶 swapF 𝐷)))
15	11, 12, 13, 14	cofuswapfcl 49278	. . 3 ⊢ (𝜑 → (𝐹 ∘func (𝐶 swapF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func 𝐸))
16		tposcurf1.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
17		tposcurf1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
18		eqid 2729	. . 3 ⊢ ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋) = ((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋)
19		tposcurf11.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20		tposcurf12.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
21		tposcurf12.1	. . 3 ⊢ 1 = (Id‘𝐶)
22		tposcurf12.y	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
23		tposcurf12.g	. . 3 ⊢ (𝜑 → 𝐻 ∈ (𝑌𝐽𝑍))
24	9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23	curf12 18133	. 2 ⊢ (𝜑 → ((𝑌(2nd ‘((1st ‘(⟨𝐶, 𝐷⟩ curryF (𝐹 ∘func (𝐶 swapF 𝐷))))‘𝑋))𝑍)‘𝐻) = (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑍⟩)𝐻))
25		eqid 2729	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
26	10, 25, 21, 11, 17	catidcl 17588	. . 3 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋(Hom ‘𝐶)𝑋))
27	11, 12, 13, 14, 10, 16, 17, 19, 17, 22, 25, 20, 26, 23	cofuswapf2 49280	. 2 ⊢ (𝜑 → (( 1 ‘𝑋)(⟨𝑋, 𝑌⟩(2nd ‘(𝐹 ∘func (𝐶 swapF 𝐷)))⟨𝑋, 𝑍⟩)𝐻) = (𝐻(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑍, 𝑋⟩)( 1 ‘𝑋)))
28	8, 24, 27	3eqtrd 2768	1 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝐻) = (𝐻(⟨𝑌, 𝑋⟩(2nd ‘𝐹)⟨𝑍, 𝑋⟩)( 1 ‘𝑋)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4583  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  Basecbs 17120  Hom chom 17172  Catccat 17570  Idccid 17571   Func cfunc 17761   ∘_func ccofu 17763   ×_c cxpc 18074   curry_F ccurf 18116   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-curf 18120  df-swapf 49245

This theorem is referenced by:  tposcurf1  49284