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Theorem mapsn 6591
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 4116 . . . 4 {𝐵} ∈ V
41, 3elmap 6578 . . 3 (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴)
5 ffn 5279 . . . . . . . 8 (𝑓:{𝐵}⟶𝐴𝑓 Fn {𝐵})
62snid 3562 . . . . . . . 8 𝐵 ∈ {𝐵}
7 fneu 5234 . . . . . . . 8 ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
85, 6, 7sylancl 410 . . . . . . 7 (𝑓:{𝐵}⟶𝐴 → ∃!𝑦 𝐵𝑓𝑦)
9 euabsn 3600 . . . . . . . 8 (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦})
10 imasng 4911 . . . . . . . . . . . 12 (𝐵 ∈ V → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
112, 10ax-mp 5 . . . . . . . . . . 11 (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦}
12 imadmrn 4898 . . . . . . . . . . . 12 (𝑓 “ dom 𝑓) = ran 𝑓
13 fdm 5285 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
1413imaeq2d 4888 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
1512, 14syl5reqr 2188 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
1611, 15syl5eqr 2187 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → {𝑦𝐵𝑓𝑦} = ran 𝑓)
1716eqeq1d 2149 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → ({𝑦𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
1817exbidv 1798 . . . . . . . 8 (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
199, 18syl5bb 191 . . . . . . 7 (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
208, 19mpbid 146 . . . . . 6 (𝑓:{𝐵}⟶𝐴 → ∃𝑦ran 𝑓 = {𝑦})
21 vex 2692 . . . . . . . . . . 11 𝑦 ∈ V
2221snid 3562 . . . . . . . . . 10 𝑦 ∈ {𝑦}
23 eleq2 2204 . . . . . . . . . 10 (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓𝑦 ∈ {𝑦}))
2422, 23mpbiri 167 . . . . . . . . 9 (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
25 frn 5288 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → ran 𝑓𝐴)
2625sseld 3100 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓𝑦𝐴))
2724, 26syl5 32 . . . . . . . 8 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑦𝐴))
28 dffn4 5358 . . . . . . . . . . . 12 (𝑓 Fn {𝐵} ↔ 𝑓:{𝐵}–onto→ran 𝑓)
295, 28sylib 121 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}–onto→ran 𝑓)
30 fof 5352 . . . . . . . . . . 11 (𝑓:{𝐵}–onto→ran 𝑓𝑓:{𝐵}⟶ran 𝑓)
3129, 30syl 14 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}⟶ran 𝑓)
32 feq3 5264 . . . . . . . . . 10 (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓𝑓:{𝐵}⟶{𝑦}))
3331, 32syl5ibcom 154 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
342, 21fsn 5599 . . . . . . . . 9 (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩})
3533, 34syl6ib 160 . . . . . . . 8 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓 = {⟨𝐵, 𝑦⟩}))
3627, 35jcad 305 . . . . . . 7 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩})))
3736eximdv 1853 . . . . . 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 2423 . . . . 5 (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
4038, 39sylibr 133 . . . 4 (𝑓:{𝐵}⟶𝐴 → ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
412, 21f1osn 5414 . . . . . . . . 9 {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}
42 f1oeq1 5363 . . . . . . . . 9 (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
4341, 42mpbiri 167 . . . . . . . 8 (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}–1-1-onto→{𝑦})
44 f1of 5374 . . . . . . . 8 (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
4543, 44syl 14 . . . . . . 7 (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶{𝑦})
46 snssi 3671 . . . . . . 7 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
47 fss 5291 . . . . . . 7 ((𝑓:{𝐵}⟶{𝑦} ∧ {𝑦} ⊆ 𝐴) → 𝑓:{𝐵}⟶𝐴)
4845, 46, 47syl2an 287 . . . . . 6 ((𝑓 = {⟨𝐵, 𝑦⟩} ∧ 𝑦𝐴) → 𝑓:{𝐵}⟶𝐴)
4948expcom 115 . . . . 5 (𝑦𝐴 → (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
5049rexlimiv 2546 . . . 4 (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴)
5140, 50impbii 125 . . 3 (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
524, 51bitri 183 . 2 (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
5352abbi2i 2255 1 (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1332  wex 1469  wcel 1481  ∃!weu 2000  {cab 2126  wrex 2418  Vcvv 2689  wss 3075  {csn 3531  cop 3534   class class class wbr 3936  dom cdm 4546  ran crn 4547  cima 4549   Fn wfn 5125  wf 5126  ontowfo 5128  1-1-ontowf1o 5129  (class class class)co 5781  𝑚 cmap 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-map 6551
This theorem is referenced by:  mapsnen  6712
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