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Theorem mapsnen 6801
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 6645 . . 3 𝑚 Fn (V × V)
2 mapsnen.1 . . 3 𝐴 ∈ V
3 mapsnen.2 . . . 4 𝐵 ∈ V
43snex 4180 . . 3 {𝐵} ∈ V
5 fnovex 5898 . . 3 (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ {𝐵} ∈ V) → (𝐴𝑚 {𝐵}) ∈ V)
61, 2, 4, 5mp3an 1337 . 2 (𝐴𝑚 {𝐵}) ∈ V
7 vex 2738 . . . 4 𝑧 ∈ V
87, 3fvex 5527 . . 3 (𝑧𝐵) ∈ V
98a1i 9 . 2 (𝑧 ∈ (𝐴𝑚 {𝐵}) → (𝑧𝐵) ∈ V)
10 vex 2738 . . . . 5 𝑤 ∈ V
113, 10opex 4223 . . . 4 𝐵, 𝑤⟩ ∈ V
1211snex 4180 . . 3 {⟨𝐵, 𝑤⟩} ∈ V
1312a1i 9 . 2 (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
142, 3mapsn 6680 . . . . . 6 (𝐴𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}}
1514abeq2i 2286 . . . . 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 2631 . . . 4 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
18 df-rex 2459 . . . 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 5506 . . . . . . . . . 10 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
21 vex 2738 . . . . . . . . . . 11 𝑦 ∈ V
223, 21fvsn 5703 . . . . . . . . . 10 ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦
2320, 22eqtrdi 2224 . . . . . . . . 9 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = 𝑦)
2423eqeq2d 2187 . . . . . . . 8 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧𝐵) ↔ 𝑤 = 𝑦))
25 equcom 1704 . . . . . . . 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 1603 . . 3 (∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
33 eleq1w 2236 . . . . 5 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
34 opeq2 3775 . . . . . . 7 (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
3534sneqd 3602 . . . . . 6 (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
3635eqeq2d 2187 . . . . 5 (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
3733, 36anbi12d 473 . . . 4 (𝑦 = 𝑤 → ((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
3810, 37ceqsexv 2774 . . 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 6760 1 (𝐴𝑚 {𝐵}) ≈ 𝐴
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wex 1490  wcel 2146  wrex 2454  Vcvv 2735  {csn 3589  cop 3592   class class class wbr 3998   × cxp 4618   Fn wfn 5203  cfv 5208  (class class class)co 5865  𝑚 cmap 6638  cen 6728
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-map 6640  df-en 6731
This theorem is referenced by: (None)
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