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Theorem mapsnd 39856
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 5045 . . . . . 6 {𝐵} ∈ V
43a1i 11 . . . . 5 (𝐵𝑊 → {𝐵} ∈ V)
52, 4syl 17 . . . 4 (𝜑 → {𝐵} ∈ V)
6 elmapg 8024 . . . 4 ((𝐴𝑉 ∧ {𝐵} ∈ V) → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
71, 5, 6syl2anc 696 . . 3 (𝜑 → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
8 ffn 6194 . . . . . . . . . . 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 4339 . . . . . . . . . . 11 (𝐵𝑊𝐵 ∈ {𝐵})
122, 11syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ {𝐵})
1312adantr 472 . . . . . . . . 9 ((𝜑𝑓:{𝐵}⟶𝐴) → 𝐵 ∈ {𝐵})
14 fneu 6144 . . . . . . . . 9 ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
1510, 13, 14syl2anc 696 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦)
16 euabsn 4393 . . . . . . . . . 10 (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦})
17 frel 6199 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → Rel 𝑓)
18 relimasn 5634 . . . . . . . . . . . . . 14 (Rel 𝑓 → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
1917, 18syl 17 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
20 imadmrn 5622 . . . . . . . . . . . . . 14 (𝑓 “ dom 𝑓) = ran 𝑓
21 fdm 6200 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
2221imaeq2d 5612 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
2320, 22syl5reqr 2797 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
2419, 23eqtr3d 2784 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → {𝑦𝐵𝑓𝑦} = ran 𝑓)
2524eqeq1d 2750 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴 → ({𝑦𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
2625exbidv 1987 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
2716, 26syl5bb 272 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
2827adantl 473 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
2915, 28mpbid 222 . . . . . . 7 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦})
30 vex 3331 . . . . . . . . . . . . . . 15 𝑦 ∈ V
3130snid 4341 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑦}
32 eleq2 2816 . . . . . . . . . . . . . 14 (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓𝑦 ∈ {𝑦}))
3331, 32mpbiri 248 . . . . . . . . . . . . 13 (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
34 frn 6202 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → ran 𝑓𝐴)
3534sseld 3731 . . . . . . . . . . . . 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 752 . . . . . . . . . 10 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦𝐴)
39 dffn4 6270 . . . . . . . . . . . . . . . 16 (𝑓 Fn {𝐵} ↔ 𝑓:{𝐵}–onto→ran 𝑓)
408, 39sylib 208 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}–onto→ran 𝑓)
41 fof 6264 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}–onto→ran 𝑓𝑓:{𝐵}⟶ran 𝑓)
4240, 41syl 17 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}⟶ran 𝑓)
43 feq3 6177 . . . . . . . . . . . . . 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 752 . . . . . . . . . . 11 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
472ad2antrr 764 . . . . . . . . . . . 12 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵𝑊)
4830a1i 11 . . . . . . . . . . . 12 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ V)
49 fsng 6555 . . . . . . . . . . . 12 ((𝐵𝑊𝑦 ∈ V) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
5047, 48, 49syl2anc 696 . . . . . . . . . . 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 555 . . . . . . . . 9 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
5352ex 449 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → (ran 𝑓 = {𝑦} → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩})))
5453eximdv 1983 . . . . . . 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 3044 . . . . . 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 6326 . . . . . . . . . . . 12 ((𝐵𝑊𝑦 ∈ V) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
612, 59, 60syl2anc 696 . . . . . . . . . . 11 (𝜑 → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
6261adantr 472 . . . . . . . . . 10 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
63 f1oeq1 6276 . . . . . . . . . . . 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 473 . . . . . . . . . 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 6286 . . . . . . . . 9 (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
6866, 67syl 17 . . . . . . . 8 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
69683adant2 1123 . . . . . . 7 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
70 snssi 4472 . . . . . . . 8 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
71703ad2ant2 1126 . . . . . . 7 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → {𝑦} ⊆ 𝐴)
72 fss 6205 . . . . . . 7 ((𝑓:{𝐵}⟶{𝑦} ∧ {𝑦} ⊆ 𝐴) → 𝑓:{𝐵}⟶𝐴)
7369, 71, 72syl2anc 696 . . . . . 6 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶𝐴)
74733exp 1112 . . . . 5 (𝜑 → (𝑦𝐴 → (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴)))
7574rexlimdv 3156 . . . 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 2868 1 (𝜑 → (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wex 1841  wcel 2127  ∃!weu 2595  {cab 2734  wrex 3039  Vcvv 3328  wss 3703  {csn 4309  cop 4315   class class class wbr 4792  dom cdm 5254  ran crn 5255  cima 5257  Rel wrel 5259   Fn wfn 6032  wf 6033  ontowfo 6035  1-1-ontowf1o 6036  (class class class)co 6801  𝑚 cmap 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-map 8013
This theorem is referenced by:  mapsnend  39859  iunmapsn  39877
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