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| Mirrors > Home > MPE Home > Th. List > ixpfn | Structured version Visualization version GIF version | ||
| Description: A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.) | 
| Ref | Expression | 
|---|---|
| ixpfn | ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fneq1 6658 | . 2 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴)) | |
| 2 | elixp2 8942 | . . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) | |
| 3 | 2 | simp2bi 1146 | . 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴) | 
| 4 | 1, 3 | vtoclga 3576 | 1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 Fn wfn 6555 ‘cfv 6560 Xcixp 8938 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 df-ixp 8939 | 
| This theorem is referenced by: ixpprc 8960 undifixp 8975 resixpfo 8977 boxcutc 8982 ixpiunwdom 9631 prdsbasfn 17517 xpsff1o 17613 sscfn1 17862 funcfn2 17915 natfn 18003 pthaus 23647 ptuncnv 23816 ptunhmeo 23817 ptcmplem2 24062 prdsbl 24505 finixpnum 37613 upixp 37737 prdstotbnd 37802 elixpconstg 45099 rrxsnicc 46320 ioorrnopnxrlem 46326 hoidmvlelem3 46617 hspdifhsp 46636 hspmbllem2 46647 | 
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