![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fnxpc | Structured version Visualization version GIF version |
Description: The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.) |
Ref | Expression |
---|---|
fnxpc | ⊢ ×c Fn (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xpc 18228 | . 2 ⊢ ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))〉(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))〉(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))〉))〉}) | |
2 | tpex 7765 | . . . 4 ⊢ {〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))〉(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))〉(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))〉))〉} ∈ V | |
3 | 2 | csbex 5317 | . . 3 ⊢ ⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))〉(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))〉(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))〉))〉} ∈ V |
4 | 3 | csbex 5317 | . 2 ⊢ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))〉(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))〉(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))〉))〉} ∈ V |
5 | 1, 4 | fnmpoi 8094 | 1 ⊢ ×c Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3478 ⦋csb 3908 {ctp 4635 〈cop 4637 × cxp 5687 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8011 2nd c2nd 8012 ndxcnx 17227 Basecbs 17245 Hom chom 17309 compcco 17310 ×c cxpc 18224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-xpc 18228 |
This theorem is referenced by: xpcbas 18234 xpchomfval 18235 xpccofval 18238 |
Copyright terms: Public domain | W3C validator |