![]() |
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 8179 | . 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | |
2 | elixp.1 | . . 3 ⊢ 𝐹 ∈ V | |
3 | 3anass 1122 | . . 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 702 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
5 | 1, 4 | bitri 267 | 1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∧ w3a 1113 ∈ wcel 2166 ∀wral 3117 Vcvv 3414 Fn wfn 6118 ‘cfv 6123 Xcixp 8175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fn 6126 df-fv 6131 df-ixp 8176 |
This theorem is referenced by: elixpconst 8183 ixpin 8200 ixpiin 8201 resixpfo 8213 elixpsn 8214 boxriin 8217 boxcutc 8218 ixpfi2 8533 ixpiunwdom 8765 dfac9 9273 ac9 9620 ac9s 9630 konigthlem 9705 xpscf 16579 cofucl 16900 yonedalem3 17273 psrbaglefi 19733 ptpjpre1 21745 ptpjcn 21785 ptpjopn 21786 ptclsg 21789 dfac14 21792 pthaus 21812 xkopt 21829 ptcmplem2 22227 ptcmplem3 22228 ptcmplem4 22229 prdsbl 22666 prdsxmslem2 22704 eulerpartlemb 30975 ptpconn 31761 finixpnum 33937 ptrest 33952 poimirlem29 33982 poimirlem30 33983 inixp 34066 prdstotbnd 34135 ioorrnopnlem 41315 hoicvr 41556 hoidmvlelem3 41605 hspdifhsp 41624 hspmbllem2 41635 |
Copyright terms: Public domain | W3C validator |