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Theorem mapsnd 38383
 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 4835 . . . . . 6 {𝐵} ∈ V
43a1i 11 . . . . 5 (𝐵𝑊 → {𝐵} ∈ V)
52, 4syl 17 . . . 4 (𝜑 → {𝐵} ∈ V)
6 elmapg 7757 . . . 4 ((𝐴𝑉 ∧ {𝐵} ∈ V) → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
71, 5, 6syl2anc 691 . . 3 (𝜑 → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴))
8 ffn 5958 . . . . . . . . . . 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 4153 . . . . . . . . . . 11 (𝐵𝑊𝐵 ∈ {𝐵})
122, 11syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ {𝐵})
1312adantr 480 . . . . . . . . 9 ((𝜑𝑓:{𝐵}⟶𝐴) → 𝐵 ∈ {𝐵})
14 fneu 5909 . . . . . . . . 9 ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
1510, 13, 14syl2anc 691 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃!𝑦 𝐵𝑓𝑦)
16 euabsn 4205 . . . . . . . . . 10 (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦})
17 frel 5963 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → Rel 𝑓)
18 relimasn 5407 . . . . . . . . . . . . . 14 (Rel 𝑓 → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
1917, 18syl 17 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
20 imadmrn 5395 . . . . . . . . . . . . . 14 (𝑓 “ dom 𝑓) = ran 𝑓
21 fdm 5964 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
2221imaeq2d 5385 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
2320, 22syl5reqr 2659 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
2419, 23eqtr3d 2646 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → {𝑦𝐵𝑓𝑦} = ran 𝑓)
2524eqeq1d 2612 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴 → ({𝑦𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
2625exbidv 1837 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → (∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦} ↔ ∃𝑦ran 𝑓 = {𝑦}))
2716, 26syl5bb 271 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
2827adantl 481 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦ran 𝑓 = {𝑦}))
2915, 28mpbid 221 . . . . . . 7 ((𝜑𝑓:{𝐵}⟶𝐴) → ∃𝑦ran 𝑓 = {𝑦})
30 vex 3176 . . . . . . . . . . . . . . 15 𝑦 ∈ V
3130snid 4155 . . . . . . . . . . . . . 14 𝑦 ∈ {𝑦}
32 eleq2 2677 . . . . . . . . . . . . . 14 (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓𝑦 ∈ {𝑦}))
3331, 32mpbiri 247 . . . . . . . . . . . . 13 (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
34 frn 5966 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴 → ran 𝑓𝐴)
3534sseld 3567 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓𝑦𝐴))
3633, 35syl5 33 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑦𝐴))
3736imp 444 . . . . . . . . . . 11 ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑦𝐴)
3837adantll 746 . . . . . . . . . 10 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦𝐴)
39 dffn4 6034 . . . . . . . . . . . . . . . 16 (𝑓 Fn {𝐵} ↔ 𝑓:{𝐵}–onto→ran 𝑓)
408, 39sylib 207 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}–onto→ran 𝑓)
41 fof 6028 . . . . . . . . . . . . . . 15 (𝑓:{𝐵}–onto→ran 𝑓𝑓:{𝐵}⟶ran 𝑓)
4240, 41syl 17 . . . . . . . . . . . . . 14 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}⟶ran 𝑓)
43 feq3 5941 . . . . . . . . . . . . . 14 (ran 𝑓 = {𝑦} → (𝑓:{𝐵}⟶ran 𝑓𝑓:{𝐵}⟶{𝑦}))
4442, 43syl5ibcom 234 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑓:{𝐵}⟶{𝑦}))
4544imp 444 . . . . . . . . . . . 12 ((𝑓:{𝐵}⟶𝐴 ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
4645adantll 746 . . . . . . . . . . 11 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓:{𝐵}⟶{𝑦})
472ad2antrr 758 . . . . . . . . . . . 12 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝐵𝑊)
4830a1i 11 . . . . . . . . . . . 12 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑦 ∈ V)
49 fsng 6310 . . . . . . . . . . . 12 ((𝐵𝑊𝑦 ∈ V) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
5047, 48, 49syl2anc 691 . . . . . . . . . . 11 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩}))
5146, 50mpbid 221 . . . . . . . . . 10 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → 𝑓 = {⟨𝐵, 𝑦⟩})
5238, 51jca 553 . . . . . . . . 9 (((𝜑𝑓:{𝐵}⟶𝐴) ∧ ran 𝑓 = {𝑦}) → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
5352ex 449 . . . . . . . 8 ((𝜑𝑓:{𝐵}⟶𝐴) → (ran 𝑓 = {𝑦} → (𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩})))
5453eximdv 1833 . . . . . . 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 2902 . . . . . 6 (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
5755, 56sylibr 223 . . . . 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 6089 . . . . . . . . . . . 12 ((𝐵𝑊𝑦 ∈ V) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
612, 59, 60syl2anc 691 . . . . . . . . . . 11 (𝜑 → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
6261adantr 480 . . . . . . . . . 10 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦})
63 f1oeq1 6040 . . . . . . . . . . . 12 (𝑓 = {⟨𝐵, 𝑦⟩} → (𝑓:{𝐵}–1-1-onto→{𝑦} ↔ {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}))
6463bicomd 212 . . . . . . . . . . 11 (𝑓 = {⟨𝐵, 𝑦⟩} → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
6564adantl 481 . . . . . . . . . 10 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → ({⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦} ↔ 𝑓:{𝐵}–1-1-onto→{𝑦}))
6662, 65mpbid 221 . . . . . . . . 9 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}–1-1-onto→{𝑦})
67 f1of 6050 . . . . . . . . 9 (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
6866, 67syl 17 . . . . . . . 8 ((𝜑𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
69683adant2 1073 . . . . . . 7 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶{𝑦})
70 snssi 4280 . . . . . . . 8 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
71703ad2ant2 1076 . . . . . . 7 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → {𝑦} ⊆ 𝐴)
72 fss 5969 . . . . . . 7 ((𝑓:{𝐵}⟶{𝑦} ∧ {𝑦} ⊆ 𝐴) → 𝑓:{𝐵}⟶𝐴)
7369, 71, 72syl2anc 691 . . . . . 6 ((𝜑𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}) → 𝑓:{𝐵}⟶𝐴)
74733exp 1256 . . . . 5 (𝜑 → (𝑦𝐴 → (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴)))
7574rexlimdv 3012 . . . 4 (𝜑 → (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶𝐴))
7658, 75impbid 201 . . 3 (𝜑 → (𝑓:{𝐵}⟶𝐴 ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
777, 76bitrd 267 . 2 (𝜑 → (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}))
7877abbi2dv 2729 1 (𝜑 → (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  {cab 2596  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ⟨cop 4131   class class class wbr 4583  dom cdm 5038  ran crn 5039   “ cima 5041  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746 This theorem is referenced by:  mapsnend  38386  iunmapsn  38404
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