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| Mirrors > Home > MPE Home > Th. List > elixp2 | Structured version Visualization version GIF version | ||
| Description: Membership in an infinite Cartesian product. See df-ixp 8938 for discussion of the notation. (Contributed by NM, 28-Sep-2006.) |
| Ref | Expression |
|---|---|
| elixp2 | ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fneq1 6659 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓 Fn 𝐴 ↔ 𝐹 Fn 𝐴)) | |
| 2 | fveq1 6905 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
| 3 | 2 | eleq1d 2826 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵)) |
| 4 | 3 | ralbidv 3178 | . . . . 5 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
| 5 | 1, 4 | anbi12d 632 | . . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))) |
| 6 | dfixp 8939 | . . . 4 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
| 7 | 5, 6 | elab2g 3680 | . . 3 ⊢ (𝐹 ∈ V → (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))) |
| 8 | 7 | pm5.32i 574 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))) |
| 9 | elex 3501 | . . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 ∈ V) | |
| 10 | 9 | pm4.71ri 560 | . 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐵)) |
| 11 | 3anass 1095 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))) | |
| 12 | 8, 10, 11 | 3bitr4i 303 | 1 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ 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: fvixp 8942 ixpfn 8943 elixp 8944 ixpf 8960 resixp 8973 undifixp 8974 mptelixpg 8975 prdsbasprj 17517 xpsfrnel 17607 xpscf 17610 isssc 17864 isfuncd 17910 funcres2b 17942 dprdw 20030 ptrecube 37627 kelac1 43075 elixpconstg 45094 fvixp2 45204 rrxsnicc 46315 ioorrnopnxrlem 46321 hoiqssbllem1 46637 iinhoiicclem 46688 iunhoiioolem 46690 funcf2lem 48914 isnatd 48949 |
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