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Theorem mapsnend 39882
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 ovexd 6835 . 2 (𝜑 → (𝐴𝑚 {𝐵}) ∈ V)
2 mapsnend.a . . 3 (𝜑𝐴𝑉)
32elexd 3346 . 2 (𝜑𝐴 ∈ V)
4 fvexd 6356 . . 3 (𝑧 ∈ (𝐴𝑚 {𝐵}) → (𝑧𝐵) ∈ V)
54a1i 11 . 2 (𝜑 → (𝑧 ∈ (𝐴𝑚 {𝐵}) → (𝑧𝐵) ∈ V))
6 snex 5049 . . . 4 {⟨𝐵, 𝑤⟩} ∈ V
76a1i 11 . . 3 (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V)
87a1i 11 . 2 (𝜑 → (𝑤𝐴 → {⟨𝐵, 𝑤⟩} ∈ V))
9 mapsnend.b . . . . . . 7 (𝜑𝐵𝑊)
102, 9mapsnd 39879 . . . . . 6 (𝜑 → (𝐴𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}})
1110abeq2d 2864 . . . . 5 (𝜑 → (𝑧 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩}))
1211anbi1d 743 . . . 4 (𝜑 → ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
13 r19.41v 3219 . . . . . 6 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
1413bicomi 214 . . . . 5 ((∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))
1514a1i 11 . . . 4 (𝜑 → ((∃𝑦𝐴 𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
16 df-rex 3048 . . . . 5 (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))))
1716a1i 11 . . . 4 (𝜑 → (∃𝑦𝐴 (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))))
1812, 15, 173bitrd 294 . . 3 (𝜑 → ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)))))
19 fveq1 6343 . . . . . . . . . . . 12 (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
2019adantl 473 . . . . . . . . . . 11 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧𝐵) = ({⟨𝐵, 𝑦⟩}‘𝐵))
21 vex 3335 . . . . . . . . . . . . . 14 𝑦 ∈ V
2221a1i 11 . . . . . . . . . . . . 13 (𝜑𝑦 ∈ V)
23 fvsng 6603 . . . . . . . . . . . . 13 ((𝐵𝑊𝑦 ∈ V) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
249, 22, 23syl2anc 696 . . . . . . . . . . . 12 (𝜑 → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
2524adantr 472 . . . . . . . . . . 11 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}‘𝐵) = 𝑦)
2620, 25eqtrd 2786 . . . . . . . . . 10 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑧𝐵) = 𝑦)
2726eqeq2d 2762 . . . . . . . . 9 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧𝐵) ↔ 𝑤 = 𝑦))
28 equcom 2092 . . . . . . . . . 10 (𝑤 = 𝑦𝑦 = 𝑤)
2928a1i 11 . . . . . . . . 9 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = 𝑦𝑦 = 𝑤))
3027, 29bitrd 268 . . . . . . . 8 ((𝜑𝑧 = {⟨𝐵, 𝑦⟩}) → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤))
3130ex 449 . . . . . . 7 (𝜑 → (𝑧 = {⟨𝐵, 𝑦⟩} → (𝑤 = (𝑧𝐵) ↔ 𝑦 = 𝑤)))
3231pm5.32d 674 . . . . . 6 (𝜑 → ((𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
3332anbi2d 742 . . . . 5 (𝜑 → ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
34 anass 684 . . . . . 6 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤)))
3534a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑦 = 𝑤))))
36 ancom 465 . . . . . 6 (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})))
3736a1i 11 . . . . 5 (𝜑 → (((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
3833, 35, 373bitr2d 296 . . . 4 (𝜑 → ((𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
3938exbidv 1991 . . 3 (𝜑 → (∃𝑦(𝑦𝐴 ∧ (𝑧 = {⟨𝐵, 𝑦⟩} ∧ 𝑤 = (𝑧𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}))))
40 vex 3335 . . . . 5 𝑤 ∈ V
41 eleq1w 2814 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
42 opeq2 4546 . . . . . . . 8 (𝑦 = 𝑤 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝑤⟩)
4342sneqd 4325 . . . . . . 7 (𝑦 = 𝑤 → {⟨𝐵, 𝑦⟩} = {⟨𝐵, 𝑤⟩})
4443eqeq2d 2762 . . . . . 6 (𝑦 = 𝑤 → (𝑧 = {⟨𝐵, 𝑦⟩} ↔ 𝑧 = {⟨𝐵, 𝑤⟩}))
4541, 44anbi12d 749 . . . . 5 (𝑦 = 𝑤 → ((𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩}) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
4640, 45ceqsexv 3374 . . . 4 (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩}))
4746a1i 11 . . 3 (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦𝐴𝑧 = {⟨𝐵, 𝑦⟩})) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
4818, 39, 473bitrd 294 . 2 (𝜑 → ((𝑧 ∈ (𝐴𝑚 {𝐵}) ∧ 𝑤 = (𝑧𝐵)) ↔ (𝑤𝐴𝑧 = {⟨𝐵, 𝑤⟩})))
491, 3, 5, 8, 48en2d 8149 1 (𝜑 → (𝐴𝑚 {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wex 1845  wcel 2131  wrex 3043  Vcvv 3332  {csn 4313  cop 4319   class class class wbr 4796  cfv 6041  (class class class)co 6805  𝑚 cmap 8015  cen 8110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-map 8017  df-en 8114
This theorem is referenced by:  mpct  39884
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