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Mirrors > Home > MPE Home > Th. List > ixpssmap | Structured version Visualization version GIF version |
Description: An infinite Cartesian product is a subset of set exponentiation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.) |
Ref | Expression |
---|---|
ixpssmap.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
ixpssmap | ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixpssmap.2 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | rgenw 3054 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V |
3 | ixpssmapg 8947 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴)) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∀wral 3050 Vcvv 3461 ⊆ wss 3944 ∪ ciun 4997 (class class class)co 7419 ↑m cmap 8845 Xcixp 8916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-map 8847 df-ixp 8917 |
This theorem is referenced by: hspmbl 46155 |
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