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Theorem mapsn 6684
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 4182 . . . 4 {𝐵} ∈ V
41, 3elmap 6671 . . 3 (𝑓 ∈ (𝐴𝑚 {𝐵}) ↔ 𝑓:{𝐵}⟶𝐴)
5 ffn 5361 . . . . . . . 8 (𝑓:{𝐵}⟶𝐴𝑓 Fn {𝐵})
62snid 3622 . . . . . . . 8 𝐵 ∈ {𝐵}
7 fneu 5316 . . . . . . . 8 ((𝑓 Fn {𝐵} ∧ 𝐵 ∈ {𝐵}) → ∃!𝑦 𝐵𝑓𝑦)
85, 6, 7sylancl 413 . . . . . . 7 (𝑓:{𝐵}⟶𝐴 → ∃!𝑦 𝐵𝑓𝑦)
9 euabsn 3661 . . . . . . . 8 (∃!𝑦 𝐵𝑓𝑦 ↔ ∃𝑦{𝑦𝐵𝑓𝑦} = {𝑦})
10 imasng 4989 . . . . . . . . . . . 12 (𝐵 ∈ V → (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦})
112, 10ax-mp 5 . . . . . . . . . . 11 (𝑓 “ {𝐵}) = {𝑦𝐵𝑓𝑦}
12 fdm 5367 . . . . . . . . . . . . 13 (𝑓:{𝐵}⟶𝐴 → dom 𝑓 = {𝐵})
1312imaeq2d 4966 . . . . . . . . . . . 12 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ {𝐵}))
14 imadmrn 4976 . . . . . . . . . . . 12 (𝑓 “ dom 𝑓) = ran 𝑓
1513, 14eqtr3di 2225 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴 → (𝑓 “ {𝐵}) = ran 𝑓)
1611, 15eqtr3id 2224 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → {𝑦𝐵𝑓𝑦} = ran 𝑓)
1716eqeq1d 2186 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → ({𝑦𝐵𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
1817exbidv 1825 . . . . . . . 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 2740 . . . . . . . . . . 11 𝑦 ∈ V
2221snid 3622 . . . . . . . . . 10 𝑦 ∈ {𝑦}
23 eleq2 2241 . . . . . . . . . 10 (ran 𝑓 = {𝑦} → (𝑦 ∈ ran 𝑓𝑦 ∈ {𝑦}))
2422, 23mpbiri 168 . . . . . . . . 9 (ran 𝑓 = {𝑦} → 𝑦 ∈ ran 𝑓)
25 frn 5370 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴 → ran 𝑓𝐴)
2625sseld 3154 . . . . . . . . 9 (𝑓:{𝐵}⟶𝐴 → (𝑦 ∈ ran 𝑓𝑦𝐴))
2724, 26syl5 32 . . . . . . . 8 (𝑓:{𝐵}⟶𝐴 → (ran 𝑓 = {𝑦} → 𝑦𝐴))
28 dffn4 5440 . . . . . . . . . . . 12 (𝑓 Fn {𝐵} ↔ 𝑓:{𝐵}–onto→ran 𝑓)
295, 28sylib 122 . . . . . . . . . . 11 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}–onto→ran 𝑓)
30 fof 5434 . . . . . . . . . . 11 (𝑓:{𝐵}–onto→ran 𝑓𝑓:{𝐵}⟶ran 𝑓)
3129, 30syl 14 . . . . . . . . . 10 (𝑓:{𝐵}⟶𝐴𝑓:{𝐵}⟶ran 𝑓)
32 feq3 5346 . . . . . . . . . 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 5684 . . . . . . . . 9 (𝑓:{𝐵}⟶{𝑦} ↔ 𝑓 = {⟨𝐵, 𝑦⟩})
3533, 34syl6ib 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 1880 . . . . . 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 2461 . . . . 5 (∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩} ↔ ∃𝑦(𝑦𝐴𝑓 = {⟨𝐵, 𝑦⟩}))
4038, 39sylibr 134 . . . 4 (𝑓:{𝐵}⟶𝐴 → ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩})
412, 21f1osn 5497 . . . . . . . . 9 {⟨𝐵, 𝑦⟩}:{𝐵}–1-1-onto→{𝑦}
42 f1oeq1 5445 . . . . . . . . 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 5457 . . . . . . . 8 (𝑓:{𝐵}–1-1-onto→{𝑦} → 𝑓:{𝐵}⟶{𝑦})
4543, 44syl 14 . . . . . . 7 (𝑓 = {⟨𝐵, 𝑦⟩} → 𝑓:{𝐵}⟶{𝑦})
46 snssi 3735 . . . . . . 7 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
47 fss 5373 . . . . . . 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 2588 . . . 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 2292 1 (𝐴𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦𝐴 𝑓 = {⟨𝐵, 𝑦⟩}}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wex 1492  ∃!weu 2026  wcel 2148  {cab 2163  wrex 2456  Vcvv 2737  wss 3129  {csn 3591  cop 3594   class class class wbr 4000  dom cdm 4623  ran crn 4624  cima 4626   Fn wfn 5207  wf 5208  ontowfo 5210  1-1-ontowf1o 5211  (class class class)co 5869  𝑚 cmap 6642
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-map 6644
This theorem is referenced by:  mapsnen  6805
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