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| 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 8941 | . 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
| 2 | elixp.1 | . . 3 ⊢ 𝐹 ∈ V | |
| 3 | 3anass 1095 | . . 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 1087 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 Fn wfn 6556 ‘cfv 6561 Xcixp 8937 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ixp 8938 |
| This theorem is referenced by: elixpconst 8945 ixpin 8963 ixpiin 8964 resixpfo 8976 elixpsn 8977 boxriin 8980 boxcutc 8981 ixpfi2 9390 ixpiunwdom 9630 dfac9 10177 ac9 10523 ac9s 10533 konigthlem 10608 cofucl 17933 yonedalem3 18325 psrbaglefi 21946 ptpjpre1 23579 ptpjcn 23619 ptpjopn 23620 ptclsg 23623 dfac14 23626 pthaus 23646 xkopt 23663 ptcmplem2 24061 ptcmplem3 24062 ptcmplem4 24063 prdsbl 24504 prdsxmslem2 24542 eulerpartlemb 34370 ptpconn 35238 finixpnum 37612 ptrest 37626 poimirlem29 37656 poimirlem30 37657 inixp 37735 prdstotbnd 37801 ioorrnopnlem 46319 hoicvr 46563 hoidmvlelem3 46612 hspdifhsp 46631 hspmbllem2 46642 |
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