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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 5893 | . 2 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴)) | |
2 | elixp2 7798 | . . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) | |
3 | 2 | simp2bi 1070 | . 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴) |
4 | 1, 3 | vtoclga 3245 | 1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 Fn wfn 5799 ‘cfv 5804 Xcixp 7794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-ixp 7795 |
This theorem is referenced by: ixpprc 7815 undifixp 7830 resixpfo 7832 boxcutc 7837 ixpiunwdom 8379 prdsbasfn 15954 xpsff1o 16051 sscfn1 16300 funcfn2 16352 natfn 16437 pthaus 21251 ptuncnv 21420 ptunhmeo 21421 ptcmplem2 21667 prdsbl 22106 finixpnum 32564 upixp 32694 prdstotbnd 32763 rrxsnicc 39196 ioorrnopnxrlem 39202 hoidmvlelem3 39487 hspdifhsp 39506 hspmbllem2 39517 |
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