xpcfuccocl - Mathbox for Zhi Wang

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Theorem xpcfuccocl 49610

Description: The composition of two natural transformations is a natural transformation. (Contributed by Zhi Wang, 1-Oct-2025.)

**Hypotheses**
Ref	Expression
xpcfuchom2.t	⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×_c (𝐷 FuncCat 𝐸))
xpcfucco2.o	⊢ 𝑂 = (comp‘𝑇)
xpcfucco2.f	⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃))
xpcfucco2.g	⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄))
xpcfucco2.k	⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅))
xpcfucco2.l	⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆))

**Assertion**
Ref	Expression
xpcfuccocl	⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) ∈ ((𝑀(𝐵 Nat 𝐶)𝑅) × (𝑁(𝐷 Nat 𝐸)𝑆)))

**Proof of Theorem xpcfuccocl**
Step	Hyp	Ref	Expression
1		xpcfuchom2.t	. . 3 ⊢ 𝑇 = ((𝐵 FuncCat 𝐶) ×_c (𝐷 FuncCat 𝐸))
2		xpcfucco2.o	. . 3 ⊢ 𝑂 = (comp‘𝑇)
3		xpcfucco2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑀(𝐵 Nat 𝐶)𝑃))
4		xpcfucco2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑁(𝐷 Nat 𝐸)𝑄))
5		xpcfucco2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝑃(𝐵 Nat 𝐶)𝑅))
6		xpcfucco2.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝑄(𝐷 Nat 𝐸)𝑆))
7	1, 2, 3, 4, 5, 6	xpcfucco2 49609	. 2 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) = ⟨(𝐾(⟨𝑀, 𝑃⟩(comp‘(𝐵 FuncCat 𝐶))𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩(comp‘(𝐷 FuncCat 𝐸))𝑆)𝐺)⟩)
8		eqid 2737	. . . 4 ⊢ (𝐵 FuncCat 𝐶) = (𝐵 FuncCat 𝐶)
9		eqid 2737	. . . 4 ⊢ (𝐵 Nat 𝐶) = (𝐵 Nat 𝐶)
10		eqid 2737	. . . 4 ⊢ (comp‘(𝐵 FuncCat 𝐶)) = (comp‘(𝐵 FuncCat 𝐶))
11	8, 9, 10, 3, 5	fuccocl 17903	. . 3 ⊢ (𝜑 → (𝐾(⟨𝑀, 𝑃⟩(comp‘(𝐵 FuncCat 𝐶))𝑅)𝐹) ∈ (𝑀(𝐵 Nat 𝐶)𝑅))
12		eqid 2737	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
13		eqid 2737	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
14		eqid 2737	. . . 4 ⊢ (comp‘(𝐷 FuncCat 𝐸)) = (comp‘(𝐷 FuncCat 𝐸))
15	12, 13, 14, 4, 6	fuccocl 17903	. . 3 ⊢ (𝜑 → (𝐿(⟨𝑁, 𝑄⟩(comp‘(𝐷 FuncCat 𝐸))𝑆)𝐺) ∈ (𝑁(𝐷 Nat 𝐸)𝑆))
16	11, 15	opelxpd 5671	. 2 ⊢ (𝜑 → ⟨(𝐾(⟨𝑀, 𝑃⟩(comp‘(𝐵 FuncCat 𝐶))𝑅)𝐹), (𝐿(⟨𝑁, 𝑄⟩(comp‘(𝐷 FuncCat 𝐸))𝑆)𝐺)⟩ ∈ ((𝑀(𝐵 Nat 𝐶)𝑅) × (𝑁(𝐷 Nat 𝐸)𝑆)))
17	7, 16	eqeltrd 2837	1 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩(⟨⟨𝑀, 𝑁⟩, ⟨𝑃, 𝑄⟩⟩𝑂⟨𝑅, 𝑆⟩)⟨𝐹, 𝐺⟩) ∈ ((𝑀(𝐵 Nat 𝐶)𝑅) × (𝑁(𝐷 Nat 𝐸)𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4588 × cxp 5630 ‘cfv 6500 (class class class)co 7368 compcco 17201 Nat cnat 17880 FuncCat cfuc 17881 ×_c cxpc 18103

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 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-hom 17213 df-cco 17214 df-cat 17603 df-func 17794 df-nat 17882 df-fuc 17883 df-xpc 18107

This theorem is referenced by: fucocolem2 49707

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