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Theorem mapsnen 6825
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.)
Hypotheses
Ref Expression
mapsnen.1 𝐴 ∈ V
mapsnen.2 𝐵 ∈ V
Assertion
Ref Expression
mapsnen (𝐴𝑚 {𝐵}) ≈ 𝐴

Proof of Theorem mapsnen
Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnmap 6669 . . 3 𝑚 Fn (V × V)
2 mapsnen.1 . . 3 𝐴 ∈ V
3 mapsnen.2 . . . 4 𝐵 ∈ V
43snex 4197 . . 3 {𝐵} ∈ V
5 fnovex 5921 . . 3 (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ {𝐵} ∈ V) → (𝐴𝑚 {𝐵}) ∈ V)
61, 2, 4, 5mp3an 1347 . 2 (𝐴𝑚 {𝐵}) ∈ V
7 vex 2752 . . . 4 𝑧 ∈ V
87, 3fvex 5547 . . 3 (𝑧𝐵) ∈ V
98a1i 9 . 2 (𝑧 ∈ (𝐴𝑚 {𝐵}) → (𝑧𝐵) ∈ V)
10 vex 2752 . . . . 5 𝑤 ∈ V
113, 10opex 4241 . . . 4 𝐵, 𝑤⟩ ∈ V
1211snex 4197 . . 3 {⟨𝐵, 𝑤⟩} ∈ V
1312a1i 9 . 2 (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
142, 3mapsn 6704 . . . . . 6 (𝐴𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}}
1514abeq2i 2298 . . . . 5 (𝑧 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩})
1615anbi1i 458 . . . 4 ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
17 r19.41v 2643 . . . 4 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
18 df-rex 2471 . . . 4 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
1916, 17, 183bitr2i 208 . . 3 ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
20 fveq1 5526 . . . . . . . . . 10 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
21 vex 2752 . . . . . . . . . . 11 𝑦 ∈ V
223, 21fvsn 5724 . . . . . . . . . 10 ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦
2320, 22eqtrdi 2236 . . . . . . . . 9 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = 𝑦)
2423eqeq2d 2199 . . . . . . . 8 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧𝐵) ↔ 𝑤 = 𝑦))
25 equcom 1716 . . . . . . . 8 (𝑤 = 𝑦𝑦 = 𝑤)
2624, 25bitrdi 196 . . . . . . 7 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤))
2726pm5.32i 454 . . . . . 6 ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))
2827anbi2i 457 . . . . 5 ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
29 anass 401 . . . . 5 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
30 ancom 266 . . . . 5 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
3128, 29, 303bitr2i 208 . . . 4 ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
3231exbii 1615 . . 3 (∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
33 eleq1w 2248 . . . . 5 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
34 opeq2 3791 . . . . . . 7 (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
3534sneqd 3617 . . . . . 6 (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
3635eqeq2d 2199 . . . . 5 (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
3733, 36anbi12d 473 . . . 4 (𝑦 = 𝑤 → ((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
3810, 37ceqsexv 2788 . . 3 (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
3919, 32, 383bitri 206 . 2 ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
406, 2, 9, 13, 39en2i 6784 1 (𝐴𝑚 {𝐵}) ≈ 𝐴
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1363  wex 1502  wcel 2158  wrex 2466  Vcvv 2749  {csn 3604  cop 3607   class class class wbr 4015   × cxp 4636   Fn wfn 5223  cfv 5228  (class class class)co 5888  𝑚 cmap 6662  cen 6752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-map 6664  df-en 6755
This theorem is referenced by: (None)
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