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Mirrors > Home > MPE Home > Th. List > ixpconstg | Structured version Visualization version GIF version |
Description: Infinite Cartesian product of a constant 𝐵. (Contributed by Mario Carneiro, 11-Jan-2015.) |
Ref | Expression |
---|---|
ixpconstg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑m 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3482 | . . . . 5 ⊢ 𝑓 ∈ V | |
2 | 1 | elixpconst 8944 | . . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝑓:𝐴⟶𝐵) |
3 | 2 | eqabi 2875 | . . 3 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
4 | mapvalg 8875 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 ↑m 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶𝐵}) | |
5 | 3, 4 | eqtr4id 2794 | . 2 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑m 𝐴)) |
6 | 5 | ancoms 458 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 ⟶wf 6559 (class class class)co 7431 ↑m cmap 8865 Xcixp 8936 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-ixp 8937 |
This theorem is referenced by: ixpconst 8946 mapsnf1o 8978 prdshom 17514 pwsbas 17534 frlmip 21816 pttoponconst 23621 xkoptsub 23678 xkopt 23679 tmdgsum2 24120 rrxip 25438 ovnlecvr2 46566 naryfvalixp 48479 |
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