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Mirrors > Home > MPE Home > Th. List > mapsnen | Structured version Visualization version GIF version |
Description: Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof shortened by AV, 17-Jul-2022.) |
Ref | Expression |
---|---|
mapsnen.1 | ⊢ 𝐴 ∈ V |
mapsnen.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
mapsnen | ⊢ (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapsnen.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | id 22 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ V) | |
3 | mapsnen.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐴 ∈ V → 𝐵 ∈ V) |
5 | 2, 4 | mapsnend 8307 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2164 Vcvv 3414 {csn 4399 class class class wbr 4875 (class class class)co 6910 ↑𝑚 cmap 8127 ≈ cen 8225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-en 8229 |
This theorem is referenced by: (None) |
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