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Theorem mapsnend 38883
Description: Set exponentiation to a singleton exponent is equinumerous to its base. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
mapsnend.a (𝜑𝐴𝑉)
mapsnend.b (𝜑𝐵𝑊)
Assertion
Ref Expression
mapsnend (𝜑 → (𝐴𝑚 {𝐵}) ≈ 𝐴)

Proof of Theorem mapsnend
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6635 . . 3 (𝐴𝑚 {𝐵}) ∈ V
21a1i 11 . 2 (𝜑 → (𝐴𝑚 {𝐵}) ∈ V)
3 mapsnend.a . . 3 (𝜑𝐴𝑉)
43elexd 3200 . 2 (𝜑𝐴 ∈ V)
5 fvex 6160 . . . 4 (𝑧𝐵) ∈ V
65a1i 11 . . 3 (𝑧 ∈ (𝐴𝑚 {𝐵}) → (𝑧𝐵) ∈ V)
76a1i 11 . 2 (𝜑 → (𝑧 ∈ (𝐴𝑚 {𝐵}) → (𝑧𝐵) ∈ V))
8 snex 4871 . . . 4 {⟨𝐵, 𝑤⟩} ∈ V
98a1i 11 . . 3 (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
109a1i 11 . 2 (𝜑 → (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V))
11 mapsnend.b . . . . . . 7 (𝜑𝐵𝑊)
123, 11mapsnd 38880 . . . . . 6 (𝜑 → (𝐴𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}})
1312abeq2d 2731 . . . . 5 (𝜑 → (𝑧 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}))
1413anbi1d 740 . . . 4 (𝜑 → ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
15 r19.41v 3081 . . . . . 6 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
1615bicomi 214 . . . . 5 ((∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
1716a1i 11 . . . 4 (𝜑 → ((∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
18 df-rex 2913 . . . . 5 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
1918a1i 11 . . . 4 (𝜑 → (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))))
2014, 17, 193bitrd 294 . . 3 (𝜑 → ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))))
21 fveq1 6149 . . . . . . . . . . . 12 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
2221adantl 482 . . . . . . . . . . 11 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
23 vex 3189 . . . . . . . . . . . . . 14 𝑦 ∈ V
2423a1i 11 . . . . . . . . . . . . 13 (𝜑𝑦 ∈ V)
25 fvsng 6404 . . . . . . . . . . . . 13 ((𝐵𝑊𝑦 ∈ V) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
2611, 24, 25syl2anc 692 . . . . . . . . . . . 12 (𝜑 → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
2726adantr 481 . . . . . . . . . . 11 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
2822, 27eqtrd 2655 . . . . . . . . . 10 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧𝐵) = 𝑦)
2928eqeq2d 2631 . . . . . . . . 9 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧𝐵) ↔ 𝑤 = 𝑦))
30 equcom 1942 . . . . . . . . . 10 (𝑤 = 𝑦𝑦 = 𝑤)
3130a1i 11 . . . . . . . . 9 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = 𝑦𝑦 = 𝑤))
3229, 31bitrd 268 . . . . . . . 8 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤))
3332ex 450 . . . . . . 7 (𝜑 → (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤)))
3433pm5.32d 670 . . . . . 6 (𝜑 → ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
3534anbi2d 739 . . . . 5 (𝜑 → ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
36 anass 680 . . . . . 6 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
3736a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
38 ancom 466 . . . . . 6 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
3938a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
4035, 37, 393bitr2d 296 . . . 4 (𝜑 → ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
4140exbidv 1847 . . 3 (𝜑 → (∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
42 vex 3189 . . . . 5 𝑤 ∈ V
43 eleq1 2686 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
44 opeq2 4373 . . . . . . . 8 (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
4544sneqd 4162 . . . . . . 7 (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
4645eqeq2d 2631 . . . . . 6 (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
4743, 46anbi12d 746 . . . . 5 (𝑦 = 𝑤 → ((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
4842, 47ceqsexv 3228 . . . 4 (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
4948a1i 11 . . 3 (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
5020, 41, 493bitrd 294 . 2 (𝜑 → ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
512, 4, 7, 10, 50en2d 7938 1 (𝜑 → (𝐴𝑚 {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  Vcvv 3186  {csn 4150  cop 4156   class class class wbr 4615  cfv 5849  (class class class)co 6607  𝑚 cmap 7805  cen 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-map 7807  df-en 7903
This theorem is referenced by:  mpct  38885
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