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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 14525 | . 2 ⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠)))))) | |
2 | ovex 7437 | . . 3 ⊢ (0..^((♯‘𝑠) + (♯‘𝑡))) ∈ V | |
3 | 2 | mptex 7219 | . 2 ⊢ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))) ∈ V |
4 | 1, 3 | fnmpoi 8052 | 1 ⊢ ++ Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Vcvv 3468 ifcif 4523 ↦ cmpt 5224 × cxp 5667 Fn wfn 6531 ‘cfv 6536 (class class class)co 7404 0cc0 11109 + caddc 11112 − cmin 11445 ..^cfzo 13630 ♯chash 14293 ++ cconcat 14524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-concat 14525 |
This theorem is referenced by: frmdplusg 18777 |
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