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Mirrors > Home > MPE Home > Th. List > mapsn | Structured version Visualization version GIF version |
Description: The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Proof shortened by AV, 17-Jul-2022.) |
Ref | Expression |
---|---|
map0.1 | ⊢ 𝐴 ∈ V |
map0.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
mapsn | ⊢ (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | map0.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | id 22 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ V) | |
3 | map0.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐴 ∈ V → 𝐵 ∈ V) |
5 | 2, 4 | mapsnd 8904 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {⟨𝐵, 𝑦⟩}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 {cab 2705 ∃wrex 3067 Vcvv 3471 {csn 4629 ⟨cop 4635 (class class class)co 7420 ↑m cmap 8844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8846 |
This theorem is referenced by: (None) |
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