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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 5979 | . 2 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴)) | |
2 | elixp2 7912 | . . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 ∈ V ∧ 𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) | |
3 | 2 | simp2bi 1077 | . 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑓 Fn 𝐴) |
4 | 1, 3 | vtoclga 3272 | 1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 Fn wfn 5883 ‘cfv 5888 Xcixp 7908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-ixp 7909 |
This theorem is referenced by: ixpprc 7929 undifixp 7944 resixpfo 7946 boxcutc 7951 ixpiunwdom 8496 prdsbasfn 16131 xpsff1o 16228 sscfn1 16477 funcfn2 16529 natfn 16614 pthaus 21441 ptuncnv 21610 ptunhmeo 21611 ptcmplem2 21857 prdsbl 22296 finixpnum 33394 upixp 33524 prdstotbnd 33593 rrxsnicc 40520 ioorrnopnxrlem 40526 hoidmvlelem3 40811 hspdifhsp 40830 hspmbllem2 40841 |
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