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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 8588 | . 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 2114 Vcvv 3494 {csn 4567 class class class wbr 5066 (class class class)co 7156 ↑m cmap 8406 ≈ cen 8506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-en 8510 |
This theorem is referenced by: (None) |
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