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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 6446 | . 2 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴)) | |
2 | elixp2 8467 | . . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) | |
3 | 2 | simp2bi 1142 | . 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 2114 ∀wral 3140 Vcvv 3496 Fn wfn 6352 ‘cfv 6357 Xcixp 8463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-ixp 8464 |
This theorem is referenced by: ixpprc 8485 undifixp 8500 resixpfo 8502 boxcutc 8507 ixpiunwdom 9057 prdsbasfn 16746 xpsff1o 16842 sscfn1 17089 funcfn2 17141 natfn 17226 pthaus 22248 ptuncnv 22417 ptunhmeo 22418 ptcmplem2 22663 prdsbl 23103 finixpnum 34879 upixp 35006 prdstotbnd 35074 elixpconstg 41362 rrxsnicc 42592 ioorrnopnxrlem 42598 hoidmvlelem3 42886 hspdifhsp 42905 hspmbllem2 42916 |
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