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Mirrors > Home > MPE Home > Th. List > ixpconst | Structured version Visualization version GIF version |
Description: Infinite Cartesian product of a constant 𝐵. (Contributed by NM, 28-Sep-2006.) |
Ref | Expression |
---|---|
ixpconst.1 | ⊢ 𝐴 ∈ V |
ixpconst.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
ixpconst | ⊢ X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑m 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpconst.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ixpconst.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | ixpconstg 8464 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑m 𝐴)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑m 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3495 (class class class)co 7150 ↑m cmap 8400 Xcixp 8455 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-ixp 8456 |
This theorem is referenced by: pwcfsdom 9999 prdsval 16722 wunfunc 17163 wunnat 17220 poimirlem30 34916 poimirlem32 34918 ovnovollem1 42931 ovnovollem2 42932 |
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