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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 6629 | . 2 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴)) | |
2 | elixp2 8878 | . . 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 3562 | 1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∀wral 3060 Vcvv 3473 Fn wfn 6527 ‘cfv 6532 Xcixp 8874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6484 df-fun 6534 df-fn 6535 df-fv 6540 df-ixp 8875 |
This theorem is referenced by: ixpprc 8896 undifixp 8911 resixpfo 8913 boxcutc 8918 ixpiunwdom 9567 prdsbasfn 17399 xpsff1o 17495 sscfn1 17746 funcfn2 17801 natfn 17887 pthaus 23071 ptuncnv 23240 ptunhmeo 23241 ptcmplem2 23486 prdsbl 23929 finixpnum 36275 upixp 36400 prdstotbnd 36465 elixpconstg 43547 rrxsnicc 44787 ioorrnopnxrlem 44793 hoidmvlelem3 45084 hspdifhsp 45103 hspmbllem2 45114 |
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