fnxpc - Metamath Proof Explorer

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Theorem fnxpc 18330

Description: The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)

**Assertion**
Ref	Expression
fnxpc	⊢ ×_c Fn (V × V)

Proof of Theorem fnxpc Dummy variables 𝑓 𝑏 𝑔 ℎ 𝑟 𝑠 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xpc 18326	. 2 ⊢ ×_c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
2		tpex 7751	. . . 4 ⊢ {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} ∈ V
3	2	csbex 5265	. . 3 ⊢ ⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} ∈ V
4	3	csbex 5265	. 2 ⊢ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} ∈ V
5	1, 4	fnmpoi 8070	1 ⊢ ×_c Fn (V × V)

Colors of variables: wff setvar class

This proof depends on syntax axioms: Vcvv 3451 ⦋csb 3847 {ctp 4588 ⟨cop 4590 × cxp 5649 Fn wfn 6526 ‘cfv 6531 (class class class)co 7412 ∈ cmpo 7414 1^st c1st 7988 2^nd c2nd 7989 ndxcnx 17351 Basecbs 17367 Hom chom 17419 compcco 17420 ×_c cxpc 18322

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7740

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-fv 6539 df-oprab 7416 df-mpo 7417 df-1st 7990 df-2nd 7991 df-xpc 18326

This theorem is used by: xpcbas 18332 xpchomfval 18333 xpccofval 18336 reldmxpc 50298

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