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Mirrors > Home > MPE Home > Th. List > ccatfn | Structured version Visualization version GIF version |
Description: The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.) (Proof shortened by AV, 29-Apr-2020.) |
Ref | Expression |
---|---|
ccatfn | ⊢ ++ Fn (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-concat 13667 | . 2 ⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠)))))) | |
2 | ovex 6956 | . . 3 ⊢ (0..^((♯‘𝑠) + (♯‘𝑡))) ∈ V | |
3 | 2 | mptex 6760 | . 2 ⊢ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))) ∈ V |
4 | 1, 3 | fnmpt2i 7521 | 1 ⊢ ++ Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3398 ifcif 4307 ↦ cmpt 4967 × cxp 5355 Fn wfn 6132 ‘cfv 6137 (class class class)co 6924 0cc0 10274 + caddc 10277 − cmin 10608 ..^cfzo 12789 ♯chash 13441 ++ cconcat 13666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-concat 13667 |
This theorem is referenced by: frmdplusg 17789 |
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