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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 14574 | . 2 ⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠)))))) | |
2 | ovex 7449 | . . 3 ⊢ (0..^((♯‘𝑠) + (♯‘𝑡))) ∈ V | |
3 | 2 | mptex 7232 | . 2 ⊢ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))) ∈ V |
4 | 1, 3 | fnmpoi 8076 | 1 ⊢ ++ Fn (V × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 Vcvv 3462 ifcif 4523 ↦ cmpt 5228 × cxp 5672 Fn wfn 6541 ‘cfv 6546 (class class class)co 7416 0cc0 11149 + caddc 11152 − cmin 11485 ..^cfzo 13675 ♯chash 14342 ++ cconcat 14573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-concat 14574 |
This theorem is referenced by: frmdplusg 18839 |
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