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Theorem mapsnd 39702
Description: The value of set exponentiation with a singleton exponent. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
mapsnd.1 (𝜑𝐴𝑉)
mapsnd.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
mapsnd (𝜑 → (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})
Distinct variable groups:   𝐴,𝑓,𝑦   𝐵,𝑓,𝑦   𝜑,𝑓,𝑦
Allowed substitution hints:   𝑉(𝑦,𝑓)   𝑊(𝑦,𝑓)

Proof of Theorem mapsnd
StepHypRef Expression
1 mapsnd.1 . . . 4 (𝜑𝐴𝑉)
2 mapsnd.2 . . . . 5 (𝜑𝐵𝑊)
3 snex 4938 . . . . . 6 {𝐵} ∈ V
43a1i 11 . . . . 5 (𝐵𝑊 → {𝐵} ∈ V)
52, 4syl 17 . . . 4 (𝜑 → {𝐵} ∈ V)
6 elmapg 7912 . . . 4 ((𝐴𝑉 ∧ {𝐵} ∈ V) → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
71, 5, 6syl2anc 694 . . 3 (𝜑 → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
8 ffn 6083 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴𝑓 Fn {𝐵})
98a1i 11 . . . . . . . . . 10 (𝜑 → (𝑓:{𝐵}⟶𝐴𝑓 Fn {𝐵}))
109imp 444 . . . . . . . . 9 ((𝜑𝑓:{𝐵}⟶𝐴) → 𝑓 Fn {𝐵})
11 snidg 4239 . . . . . . . . . . 11 (𝐵𝑊𝐵 ∈ {𝐵})
122, 11syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ {𝐵})
1312adantr 480 . . . . . . . . 9 ((𝜑𝑓:{𝐵}⟶𝐴) → 𝐵 ∈ {𝐵})
14 fneu 6033 . . . . . . . . 9 ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
1510, 13, 14syl2anc 694 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦)
16 euabsn 4293 . . . . . . . . . 10 (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦})
17 frel 6088 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → Rel 𝑓)
18 relimasn 5523 . . . . . . . . . . . . . 14 (Rel 𝑓 → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
1917, 18syl 17 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
20 imadmrn 5511 . . . . . . . . . . . . . 14 (𝑓 “ dom 𝑓) = ran 𝑓
21 fdm 6089 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
2221imaeq2d 5501 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
2320, 22syl5reqr 2700 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
2419, 23eqtr3d 2687 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → {𝑦𝐵𝑓𝑦} = ran 𝑓)
2524eqeq1d 2653 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴 → ({𝑦𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
2625exbidv 1890 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
2716, 26syl5bb 272 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
2827adantl 481 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
2915, 28mpbid 222 . . . . . . 7 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦})
30 vex 3234 . . . . . . . . . . . . . . 15 𝑦 ∈ V
3130snid 4241 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑦}
32 eleq2 2719 . . . . . . . . . . . . . 14 (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓𝑦 ∈ {𝑦}))
3331, 32mpbiri 248 . . . . . . . . . . . . 13 (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
34 frn 6091 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → ran 𝑓𝐴)
3534sseld 3635 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓𝑦𝐴))
3633, 35syl5 34 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑦𝐴))
3736imp 444 . . . . . . . . . . 11 ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑦𝐴)
3837adantll 750 . . . . . . . . . 10 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦𝐴)
39 dffn4 6159 . . . . . . . . . . . . . . . 16 (𝑓 Fn {𝐵} ↔ 𝑓:{𝐵}–onto→ran 𝑓)
408, 39sylib 208 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}–onto→ran 𝑓)
41 fof 6153 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}–onto→ran 𝑓𝑓:{𝐵}⟶ran 𝑓)
4240, 41syl 17 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}⟶ran 𝑓)
43 feq3 6066 . . . . . . . . . . . . . 14 (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓𝑓:{𝐵}⟶{𝑦}))
4442, 43syl5ibcom 235 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
4544imp 444 . . . . . . . . . . . 12 ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
4645adantll 750 . . . . . . . . . . 11 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
472ad2antrr 762 . . . . . . . . . . . 12 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵𝑊)
4830a1i 11 . . . . . . . . . . . 12 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ V)
49 fsng 6444 . . . . . . . . . . . 12 ((𝐵𝑊𝑦 ∈ V) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
5047, 48, 49syl2anc 694 . . . . . . . . . . 11 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
5146, 50mpbid 222 . . . . . . . . . 10 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓 = {⟨𝐵, 𝑦⟩})
5238, 51jca 553 . . . . . . . . 9 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
5352ex 449 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → (ran 𝑓 = {𝑦} → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩})))
5453eximdv 1886 . . . . . . 7 ((𝜑𝑓:{𝐵}⟶𝐴) → (∃𝑦ran 𝑓 = {𝑦} → ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩})))
5529, 54mpd 15 . . . . . 6 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
56 df-rex 2947 . . . . . 6 (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
5755, 56sylibr 224 . . . . 5 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
5857ex 449 . . . 4 (𝜑 → (𝑓:{𝐵}⟶𝐴 → ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
5930a1i 11 . . . . . . . . . . . 12 (𝜑𝑦 ∈ V)
60 f1osng 6215 . . . . . . . . . . . 12 ((𝐵𝑊𝑦 ∈ V) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
612, 59, 60syl2anc 694 . . . . . . . . . . 11 (𝜑 → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
6261adantr 480 . . . . . . . . . 10 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
63 f1oeq1 6165 . . . . . . . . . . . 12 (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
6463bicomd 213 . . . . . . . . . . 11 (𝑓 = {⟨𝐵, 𝑦⟩} → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
6564adantl 481 . . . . . . . . . 10 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
6662, 65mpbid 222 . . . . . . . . 9 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}–1-1-onto→{𝑦})
67 f1of 6175 . . . . . . . . 9 (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
6866, 67syl 17 . . . . . . . 8 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
69683adant2 1100 . . . . . . 7 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
70 snssi 4371 . . . . . . . 8 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
71703ad2ant2 1103 . . . . . . 7 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → {𝑦} ⊆ 𝐴)
72 fss 6094 . . . . . . 7 ((𝑓:{𝐵}⟶{𝑦} ∧ {𝑦} ⊆ 𝐴) → 𝑓:{𝐵}⟶𝐴)
7369, 71, 72syl2anc 694 . . . . . 6 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶𝐴)
74733exp 1283 . . . . 5 (𝜑 → (𝑦𝐴 → (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴)))
7574rexlimdv 3059 . . . 4 (𝜑 → (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
7658, 75impbid 202 . . 3 (𝜑 → (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
777, 76bitrd 268 . 2 (𝜑 → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
7877abbi2dv 2771 1 (𝜑 → (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  {cab 2637  wrex 2942  Vcvv 3231  wss 3607  {csn 4210  cop 4216   class class class wbr 4685  dom cdm 5143  ran crn 5144  cima 5146  Rel wrel 5148   Fn wfn 5921  wf 5922  ontowfo 5924  1-1-ontowf1o 5925  (class class class)co 6690  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901
This theorem is referenced by:  mapsnend  39705  iunmapsn  39723
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