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Theorem mapsn 6715
Description: The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.)
Hypotheses
Ref Expression
map0.1 𝐴 ∈ V
map0.2 𝐵 ∈ V
Assertion
Ref Expression
mapsn (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}
Distinct variable groups:   𝑦,𝑓,𝐴   𝐵,𝑓,𝑦

Proof of Theorem mapsn
StepHypRef Expression
1 map0.1 . . . 4 𝐴 ∈ V
2 map0.2 . . . . 5 𝐵 ∈ V
32snex 4203 . . . 4 {𝐵} ∈ V
41, 3elmap 6702 . . 3 (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴)
5 ffn 5384 . . . . . . . 8 (𝑓:{𝐵}⟶𝐴𝑓 Fn {𝐵})
62snid 3638 . . . . . . . 8 𝐵 ∈ {𝐵}
7 fneu 5339 . . . . . . . 8 ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
85, 6, 7sylancl 413 . . . . . . 7 (𝑓:{𝐵}⟶𝐴 → ∃!𝑦 𝐵𝑓𝑦)
9 euabsn 3677 . . . . . . . 8 (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦})
10 imasng 5011 . . . . . . . . . . . 12 (𝐵 ∈ V → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
112, 10ax-mp 5 . . . . . . . . . . 11 (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦}
12 fdm 5390 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
1312imaeq2d 4988 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
14 imadmrn 4998 . . . . . . . . . . . 12 (𝑓 “ dom 𝑓) = ran 𝑓
1513, 14eqtr3di 2237 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
1611, 15eqtr3id 2236 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → {𝑦𝐵𝑓𝑦} = ran 𝑓)
1716eqeq1d 2198 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → ({𝑦𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
1817exbidv 1836 . . . . . . . 8 (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
199, 18bitrid 192 . . . . . . 7 (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
208, 19mpbid 147 . . . . . 6 (𝑓:{𝐵}⟶𝐴 → ∃𝑦ran 𝑓 = {𝑦})
21 vex 2755 . . . . . . . . . . 11 𝑦 ∈ V
2221snid 3638 . . . . . . . . . 10 𝑦 ∈ {𝑦}
23 eleq2 2253 . . . . . . . . . 10 (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓𝑦 ∈ {𝑦}))
2422, 23mpbiri 168 . . . . . . . . 9 (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
25 frn 5393 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → ran 𝑓𝐴)
2625sseld 3169 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓𝑦𝐴))
2724, 26syl5 32 . . . . . . . 8 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑦𝐴))
28 dffn4 5463 . . . . . . . . . . . 12 (𝑓 Fn {𝐵} ↔ 𝑓:{𝐵}–onto→ran 𝑓)
295, 28sylib 122 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}–onto→ran 𝑓)
30 fof 5457 . . . . . . . . . . 11 (𝑓:{𝐵}–onto→ran 𝑓𝑓:{𝐵}⟶ran 𝑓)
3129, 30syl 14 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}⟶ran 𝑓)
32 feq3 5369 . . . . . . . . . 10 (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓𝑓:{𝐵}⟶{𝑦}))
3331, 32syl5ibcom 155 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
342, 21fsn 5708 . . . . . . . . 9 (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩})
3533, 34imbitrdi 161 . . . . . . . 8 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓 = {⟨𝐵, 𝑦⟩}))
3627, 35jcad 307 . . . . . . 7 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩})))
3736eximdv 1891 . . . . . 6 (𝑓:{𝐵}⟶𝐴 → (∃𝑦ran 𝑓 = {𝑦} → ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩})))
3820, 37mpd 13 . . . . 5 (𝑓:{𝐵}⟶𝐴 → ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
39 df-rex 2474 . . . . 5 (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
4038, 39sylibr 134 . . . 4 (𝑓:{𝐵}⟶𝐴 → ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
412, 21f1osn 5520 . . . . . . . . 9 {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}
42 f1oeq1 5468 . . . . . . . . 9 (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
4341, 42mpbiri 168 . . . . . . . 8 (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}–1-1-onto→{𝑦})
44 f1of 5480 . . . . . . . 8 (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
4543, 44syl 14 . . . . . . 7 (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶{𝑦})
46 snssi 3751 . . . . . . 7 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
47 fss 5396 . . . . . . 7 ((𝑓:{𝐵}⟶{𝑦} ∧ {𝑦} ⊆ 𝐴) → 𝑓:{𝐵}⟶𝐴)
4845, 46, 47syl2an 289 . . . . . 6 ((𝑓 = {⟨𝐵, 𝑦⟩} ∧ 𝑦𝐴) → 𝑓:{𝐵}⟶𝐴)
4948expcom 116 . . . . 5 (𝑦𝐴 → (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
5049rexlimiv 2601 . . . 4 (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴)
5140, 50impbii 126 . . 3 (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
524, 51bitri 184 . 2 (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
5352abbi2i 2304 1 (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wex 1503  ∃!weu 2038  wcel 2160  {cab 2175  wrex 2469  Vcvv 2752  wss 3144  {csn 3607  cop 3610   class class class wbr 4018  dom cdm 4644  ran crn 4645  cima 4647   Fn wfn 5230  wf 5231  ontowfo 5233  1-1-ontowf1o 5234  (class class class)co 5895  𝑚 cmap 6673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-map 6675
This theorem is referenced by:  mapsnen  6836
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