![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elixp | Structured version Visualization version GIF version |
Description: Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.) |
Ref | Expression |
---|---|
elixp.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
elixp | ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elixp2 8940 | . 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | elixp.1 | . . 3 ⊢ 𝐹 ∈ V | |
3 | 3anass 1094 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))) | |
4 | 2, 3 | mpbiran 709 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
5 | 1, 4 | bitri 275 | 1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 Fn wfn 6558 ‘cfv 6563 Xcixp 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-ixp 8937 |
This theorem is referenced by: elixpconst 8944 ixpin 8962 ixpiin 8963 resixpfo 8975 elixpsn 8976 boxriin 8979 boxcutc 8980 ixpfi2 9388 ixpiunwdom 9628 dfac9 10175 ac9 10521 ac9s 10531 konigthlem 10606 cofucl 17939 yonedalem3 18337 psrbaglefi 21964 ptpjpre1 23595 ptpjcn 23635 ptpjopn 23636 ptclsg 23639 dfac14 23642 pthaus 23662 xkopt 23679 ptcmplem2 24077 ptcmplem3 24078 ptcmplem4 24079 prdsbl 24520 prdsxmslem2 24558 eulerpartlemb 34350 ptpconn 35218 finixpnum 37592 ptrest 37606 poimirlem29 37636 poimirlem30 37637 inixp 37715 prdstotbnd 37781 ioorrnopnlem 46260 hoicvr 46504 hoidmvlelem3 46553 hspdifhsp 46572 hspmbllem2 46583 |
Copyright terms: Public domain | W3C validator |