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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 7798 | . 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | elixp.1 | . . 3 ⊢ 𝐹 ∈ V | |
3 | 3anass 1035 | . . 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 955 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
5 | 1, 4 | bitri 263 | 1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∈ 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: elixpconst 7802 ixpin 7819 ixpiin 7820 resixpfo 7832 elixpsn 7833 boxriin 7836 boxcutc 7837 ixpfi2 8147 ixpiunwdom 8379 dfac9 8841 ac9 9188 ac9s 9198 konigthlem 9269 xpscf 16049 cofucl 16371 yonedalem3 16743 psrbaglefi 19193 ptpjpre1 21184 ptpjcn 21224 ptpjopn 21225 ptclsg 21228 dfac14 21231 pthaus 21251 xkopt 21268 ptcmplem2 21667 ptcmplem3 21668 ptcmplem4 21669 prdsbl 22106 prdsxmslem2 22144 eulerpartlemb 29757 ptpcon 30469 finixpnum 32564 ptrest 32578 poimirlem29 32608 poimirlem30 32609 inixp 32693 prdstotbnd 32763 ioorrnopnlem 39200 hoicvr 39438 hoidmvlelem3 39487 hspdifhsp 39506 hspmbllem2 39517 |
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