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Theorem mapsnd 8444
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 snex 5328 . . . . 5 {𝐵} ∈ V
32a1i 11 . . . 4 (𝜑 → {𝐵} ∈ V)
41, 3elmapd 8415 . . 3 (𝜑 → (𝑓 ∈ (𝐴m {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
5 ffn 6513 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴𝑓 Fn {𝐵})
6 mapsnd.2 . . . . . . . . . 10 (𝜑𝐵𝑊)
7 snidg 4596 . . . . . . . . . 10 (𝐵𝑊𝐵 ∈ {𝐵})
86, 7syl 17 . . . . . . . . 9 (𝜑𝐵 ∈ {𝐵})
9 fneu 6460 . . . . . . . . 9 ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
105, 8, 9syl2anr 596 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦)
11 euabsn 4661 . . . . . . . . . 10 (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦})
12 frel 6518 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → Rel 𝑓)
13 relimasn 5951 . . . . . . . . . . . . . 14 (Rel 𝑓 → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
1412, 13syl 17 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
15 imadmrn 5938 . . . . . . . . . . . . . 14 (𝑓 “ dom 𝑓) = ran 𝑓
16 fdm 6521 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
1716imaeq2d 5928 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
1815, 17syl5reqr 2876 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
1914, 18eqtr3d 2863 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → {𝑦𝐵𝑓𝑦} = ran 𝑓)
2019eqeq1d 2828 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴 → ({𝑦𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
2120exbidv 1915 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
2211, 21syl5bb 284 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
2322adantl 482 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
2410, 23mpbid 233 . . . . . . 7 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦})
25 frn 6519 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → ran 𝑓𝐴)
2625sseld 3970 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓𝑦𝐴))
27 vsnid 4599 . . . . . . . . . . . . 13 𝑦 ∈ {𝑦}
28 eleq2 2906 . . . . . . . . . . . . 13 (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓𝑦 ∈ {𝑦}))
2927, 28mpbiri 259 . . . . . . . . . . . 12 (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
3026, 29impel 506 . . . . . . . . . . 11 ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑦𝐴)
3130adantll 710 . . . . . . . . . 10 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦𝐴)
32 ffrn 6525 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}⟶ran 𝑓)
33 feq3 6496 . . . . . . . . . . . . . 14 (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓𝑓:{𝐵}⟶{𝑦}))
3432, 33syl5ibcom 246 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
3534imp 407 . . . . . . . . . . . 12 ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
3635adantll 710 . . . . . . . . . . 11 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
376ad2antrr 722 . . . . . . . . . . . 12 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵𝑊)
38 vex 3503 . . . . . . . . . . . 12 𝑦 ∈ V
39 fsng 6897 . . . . . . . . . . . 12 ((𝐵𝑊𝑦 ∈ V) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
4037, 38, 39sylancl 586 . . . . . . . . . . 11 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
4136, 40mpbid 233 . . . . . . . . . 10 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓 = {⟨𝐵, 𝑦⟩})
4231, 41jca 512 . . . . . . . . 9 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
4342ex 413 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → (ran 𝑓 = {𝑦} → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩})))
4443eximdv 1911 . . . . . . 7 ((𝜑𝑓:{𝐵}⟶𝐴) → (∃𝑦ran 𝑓 = {𝑦} → ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩})))
4524, 44mpd 15 . . . . . 6 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
46 df-rex 3149 . . . . . 6 (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
4745, 46sylibr 235 . . . . 5 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
4847ex 413 . . . 4 (𝜑 → (𝑓:{𝐵}⟶𝐴 → ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
49 f1osng 6654 . . . . . . . . . . 11 ((𝐵𝑊𝑦 ∈ V) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
506, 38, 49sylancl 586 . . . . . . . . . 10 (𝜑 → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
5150adantr 481 . . . . . . . . 9 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
52 f1oeq1 6603 . . . . . . . . . . 11 (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
5352bicomd 224 . . . . . . . . . 10 (𝑓 = {⟨𝐵, 𝑦⟩} → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
5453adantl 482 . . . . . . . . 9 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
5551, 54mpbid 233 . . . . . . . 8 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}–1-1-onto→{𝑦})
56 f1of 6614 . . . . . . . 8 (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
5755, 56syl 17 . . . . . . 7 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
58573adant2 1125 . . . . . 6 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
59 snssi 4740 . . . . . . 7 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
60593ad2ant2 1128 . . . . . 6 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → {𝑦} ⊆ 𝐴)
6158, 60fssd 6527 . . . . 5 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶𝐴)
6261rexlimdv3a 3291 . . . 4 (𝜑 → (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
6348, 62impbid 213 . . 3 (𝜑 → (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
644, 63bitrd 280 . 2 (𝜑 → (𝑓 ∈ (𝐴m {𝐵}) ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
6564abbi2dv 2955 1 (𝜑 → (𝐴m {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  ∃!weu 2651  {cab 2804  wrex 3144  Vcvv 3500  wss 3940  {csn 4564  cop 4570   class class class wbr 5063  dom cdm 5554  ran crn 5555  cima 5557  Rel wrel 5559   Fn wfn 6349  wf 6350  1-1-ontowf1o 6353  (class class class)co 7150  m cmap 8401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8403
This theorem is referenced by:  mapsn  8446  mapsnend  8582  iunmapsn  41364
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