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Theorem mapsnd 8102
Description: The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised 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 5064 . . . . . 6 {𝐵} ∈ V
43a1i 11 . . . . 5 (𝐵𝑊 → {𝐵} ∈ V)
52, 4syl 17 . . . 4 (𝜑 → {𝐵} ∈ V)
6 elmapg 8073 . . . 4 ((𝐴𝑉 ∧ {𝐵} ∈ V) → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
71, 5, 6syl2anc 579 . . 3 (𝜑 → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
8 ffn 6223 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴𝑓 Fn {𝐵})
98a1i 11 . . . . . . . . . 10 (𝜑 → (𝑓:{𝐵}⟶𝐴𝑓 Fn {𝐵}))
109imp 395 . . . . . . . . 9 ((𝜑𝑓:{𝐵}⟶𝐴) → 𝑓 Fn {𝐵})
11 snidg 4364 . . . . . . . . . . 11 (𝐵𝑊𝐵 ∈ {𝐵})
122, 11syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ {𝐵})
1312adantr 472 . . . . . . . . 9 ((𝜑𝑓:{𝐵}⟶𝐴) → 𝐵 ∈ {𝐵})
14 fneu 6173 . . . . . . . . 9 ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
1510, 13, 14syl2anc 579 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦)
16 euabsn 4416 . . . . . . . . . 10 (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦})
17 frel 6228 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → Rel 𝑓)
18 relimasn 5670 . . . . . . . . . . . . . 14 (Rel 𝑓 → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
1917, 18syl 17 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
20 imadmrn 5658 . . . . . . . . . . . . . 14 (𝑓 “ dom 𝑓) = ran 𝑓
21 fdm 6231 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
2221imaeq2d 5648 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
2320, 22syl5reqr 2814 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
2419, 23eqtr3d 2801 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → {𝑦𝐵𝑓𝑦} = ran 𝑓)
2524eqeq1d 2767 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴 → ({𝑦𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
2625exbidv 2016 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
2716, 26syl5bb 274 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
2827adantl 473 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
2915, 28mpbid 223 . . . . . . 7 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦})
30 vex 3353 . . . . . . . . . . . . . . 15 𝑦 ∈ V
3130snid 4366 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑦}
32 eleq2 2833 . . . . . . . . . . . . . 14 (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓𝑦 ∈ {𝑦}))
3331, 32mpbiri 249 . . . . . . . . . . . . 13 (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
34 frn 6229 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → ran 𝑓𝐴)
3534sseld 3760 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓𝑦𝐴))
3633, 35syl5 34 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑦𝐴))
3736imp 395 . . . . . . . . . . 11 ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑦𝐴)
3837adantll 705 . . . . . . . . . 10 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦𝐴)
39 dffn4 6304 . . . . . . . . . . . . . . . 16 (𝑓 Fn {𝐵} ↔ 𝑓:{𝐵}–onto→ran 𝑓)
408, 39sylib 209 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}–onto→ran 𝑓)
41 fof 6298 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}–onto→ran 𝑓𝑓:{𝐵}⟶ran 𝑓)
4240, 41syl 17 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}⟶ran 𝑓)
43 feq3 6206 . . . . . . . . . . . . . 14 (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓𝑓:{𝐵}⟶{𝑦}))
4442, 43syl5ibcom 236 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
4544imp 395 . . . . . . . . . . . 12 ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
4645adantll 705 . . . . . . . . . . 11 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
472ad2antrr 717 . . . . . . . . . . . 12 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵𝑊)
4830a1i 11 . . . . . . . . . . . 12 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ V)
49 fsng 6595 . . . . . . . . . . . 12 ((𝐵𝑊𝑦 ∈ V) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
5047, 48, 49syl2anc 579 . . . . . . . . . . 11 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
5146, 50mpbid 223 . . . . . . . . . 10 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓 = {⟨𝐵, 𝑦⟩})
5238, 51jca 507 . . . . . . . . 9 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
5352ex 401 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → (ran 𝑓 = {𝑦} → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩})))
5453eximdv 2012 . . . . . . 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 3061 . . . . . 6 (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
5755, 56sylibr 225 . . . . 5 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
5857ex 401 . . . 4 (𝜑 → (𝑓:{𝐵}⟶𝐴 → ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
5930a1i 11 . . . . . . . . . . . 12 (𝜑𝑦 ∈ V)
60 f1osng 6360 . . . . . . . . . . . 12 ((𝐵𝑊𝑦 ∈ V) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
612, 59, 60syl2anc 579 . . . . . . . . . . 11 (𝜑 → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
6261adantr 472 . . . . . . . . . 10 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
63 f1oeq1 6310 . . . . . . . . . . . 12 (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
6463bicomd 214 . . . . . . . . . . 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 223 . . . . . . . . 9 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}–1-1-onto→{𝑦})
67 f1of 6320 . . . . . . . . 9 (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
6866, 67syl 17 . . . . . . . 8 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
69683adant2 1161 . . . . . . 7 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
70 snssi 4493 . . . . . . . 8 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
71703ad2ant2 1164 . . . . . . 7 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → {𝑦} ⊆ 𝐴)
72 fss 6236 . . . . . . 7 ((𝑓:{𝐵}⟶{𝑦} ∧ {𝑦} ⊆ 𝐴) → 𝑓:{𝐵}⟶𝐴)
7369, 71, 72syl2anc 579 . . . . . 6 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶𝐴)
74733exp 1148 . . . . 5 (𝜑 → (𝑦𝐴 → (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴)))
7574rexlimdv 3177 . . . 4 (𝜑 → (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
7658, 75impbid 203 . . 3 (𝜑 → (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
777, 76bitrd 270 . 2 (𝜑 → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
7877abbi2dv 2885 1 (𝜑 → (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  ∃!weu 2581  {cab 2751  wrex 3056  Vcvv 3350  wss 3732  {csn 4334  cop 4340   class class class wbr 4809  dom cdm 5277  ran crn 5278  cima 5280  Rel wrel 5282   Fn wfn 6063  wf 6064  ontowfo 6066  1-1-ontowf1o 6067  (class class class)co 6842  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-map 8062
This theorem is referenced by:  mapsn  8104  mapsnend  8239  iunmapsn  40054
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