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Theorem mapsnd 8827
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 (πœ‘ β†’ 𝐴 ∈ 𝑉)
mapsnd.2 (πœ‘ β†’ 𝐡 ∈ π‘Š)
Assertion
Ref Expression
mapsnd (πœ‘ β†’ (𝐴 ↑m {𝐡}) = {𝑓 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑓 = {⟨𝐡, π‘¦βŸ©}})
Distinct variable groups:   𝐴,𝑓,𝑦   𝐡,𝑓,𝑦   πœ‘,𝑓,𝑦
Allowed substitution hints:   𝑉(𝑦,𝑓)   π‘Š(𝑦,𝑓)

Proof of Theorem mapsnd
StepHypRef Expression
1 mapsnd.1 . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑉)
2 snex 5389 . . . . 5 {𝐡} ∈ V
32a1i 11 . . . 4 (πœ‘ β†’ {𝐡} ∈ V)
41, 3elmapd 8782 . . 3 (πœ‘ β†’ (𝑓 ∈ (𝐴 ↑m {𝐡}) ↔ 𝑓:{𝐡}⟢𝐴))
5 ffn 6669 . . . . . . . . 9 (𝑓:{𝐡}⟢𝐴 β†’ 𝑓 Fn {𝐡})
6 mapsnd.2 . . . . . . . . . 10 (πœ‘ β†’ 𝐡 ∈ π‘Š)
7 snidg 4621 . . . . . . . . . 10 (𝐡 ∈ π‘Š β†’ 𝐡 ∈ {𝐡})
86, 7syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐡 ∈ {𝐡})
9 fneu 6613 . . . . . . . . 9 ((𝑓 Fn {𝐡} ∧ 𝐡 ∈ {𝐡}) β†’ βˆƒ!𝑦 𝐡𝑓𝑦)
105, 8, 9syl2anr 598 . . . . . . . 8 ((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) β†’ βˆƒ!𝑦 𝐡𝑓𝑦)
11 euabsn 4688 . . . . . . . . . 10 (βˆƒ!𝑦 𝐡𝑓𝑦 ↔ βˆƒπ‘¦{𝑦 ∣ 𝐡𝑓𝑦} = {𝑦})
12 frel 6674 . . . . . . . . . . . . . 14 (𝑓:{𝐡}⟢𝐴 β†’ Rel 𝑓)
13 relimasn 6037 . . . . . . . . . . . . . 14 (Rel 𝑓 β†’ (𝑓 β€œ {𝐡}) = {𝑦 ∣ 𝐡𝑓𝑦})
1412, 13syl 17 . . . . . . . . . . . . 13 (𝑓:{𝐡}⟢𝐴 β†’ (𝑓 β€œ {𝐡}) = {𝑦 ∣ 𝐡𝑓𝑦})
15 fdm 6678 . . . . . . . . . . . . . . 15 (𝑓:{𝐡}⟢𝐴 β†’ dom 𝑓 = {𝐡})
1615imaeq2d 6014 . . . . . . . . . . . . . 14 (𝑓:{𝐡}⟢𝐴 β†’ (𝑓 β€œ dom 𝑓) = (𝑓 β€œ {𝐡}))
17 imadmrn 6024 . . . . . . . . . . . . . 14 (𝑓 β€œ dom 𝑓) = ran 𝑓
1816, 17eqtr3di 2788 . . . . . . . . . . . . 13 (𝑓:{𝐡}⟢𝐴 β†’ (𝑓 β€œ {𝐡}) = ran 𝑓)
1914, 18eqtr3d 2775 . . . . . . . . . . . 12 (𝑓:{𝐡}⟢𝐴 β†’ {𝑦 ∣ 𝐡𝑓𝑦} = ran 𝑓)
2019eqeq1d 2735 . . . . . . . . . . 11 (𝑓:{𝐡}⟢𝐴 β†’ ({𝑦 ∣ 𝐡𝑓𝑦} = {𝑦} ↔ ran 𝑓 = {𝑦}))
2120exbidv 1925 . . . . . . . . . 10 (𝑓:{𝐡}⟢𝐴 β†’ (βˆƒπ‘¦{𝑦 ∣ 𝐡𝑓𝑦} = {𝑦} ↔ βˆƒπ‘¦ran 𝑓 = {𝑦}))
2211, 21bitrid 283 . . . . . . . . 9 (𝑓:{𝐡}⟢𝐴 β†’ (βˆƒ!𝑦 𝐡𝑓𝑦 ↔ βˆƒπ‘¦ran 𝑓 = {𝑦}))
2322adantl 483 . . . . . . . 8 ((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) β†’ (βˆƒ!𝑦 𝐡𝑓𝑦 ↔ βˆƒπ‘¦ran 𝑓 = {𝑦}))
2410, 23mpbid 231 . . . . . . 7 ((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) β†’ βˆƒπ‘¦ran 𝑓 = {𝑦})
25 frn 6676 . . . . . . . . . . . . 13 (𝑓:{𝐡}⟢𝐴 β†’ ran 𝑓 βŠ† 𝐴)
2625sseld 3944 . . . . . . . . . . . 12 (𝑓:{𝐡}⟢𝐴 β†’ (𝑦 ∈ ran 𝑓 β†’ 𝑦 ∈ 𝐴))
27 vsnid 4624 . . . . . . . . . . . . 13 𝑦 ∈ {𝑦}
28 eleq2 2823 . . . . . . . . . . . . 13 (ran 𝑓 = {𝑦} β†’ (𝑦 ∈ ran 𝑓 ↔ 𝑦 ∈ {𝑦}))
2927, 28mpbiri 258 . . . . . . . . . . . 12 (ran 𝑓 = {𝑦} β†’ 𝑦 ∈ ran 𝑓)
3026, 29impel 507 . . . . . . . . . . 11 ((𝑓:{𝐡}⟢𝐴 ∧ ran 𝑓 = {𝑦}) β†’ 𝑦 ∈ 𝐴)
3130adantll 713 . . . . . . . . . 10 (((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) ∧ ran 𝑓 = {𝑦}) β†’ 𝑦 ∈ 𝐴)
32 ffrn 6683 . . . . . . . . . . . . . 14 (𝑓:{𝐡}⟢𝐴 β†’ 𝑓:{𝐡}⟢ran 𝑓)
33 feq3 6652 . . . . . . . . . . . . . 14 (ran 𝑓 = {𝑦} β†’ (𝑓:{𝐡}⟢ran 𝑓 ↔ 𝑓:{𝐡}⟢{𝑦}))
3432, 33syl5ibcom 244 . . . . . . . . . . . . 13 (𝑓:{𝐡}⟢𝐴 β†’ (ran 𝑓 = {𝑦} β†’ 𝑓:{𝐡}⟢{𝑦}))
3534imp 408 . . . . . . . . . . . 12 ((𝑓:{𝐡}⟢𝐴 ∧ ran 𝑓 = {𝑦}) β†’ 𝑓:{𝐡}⟢{𝑦})
3635adantll 713 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) ∧ ran 𝑓 = {𝑦}) β†’ 𝑓:{𝐡}⟢{𝑦})
376ad2antrr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) ∧ ran 𝑓 = {𝑦}) β†’ 𝐡 ∈ π‘Š)
38 vex 3448 . . . . . . . . . . . 12 𝑦 ∈ V
39 fsng 7084 . . . . . . . . . . . 12 ((𝐡 ∈ π‘Š ∧ 𝑦 ∈ V) β†’ (𝑓:{𝐡}⟢{𝑦} ↔ 𝑓 = {⟨𝐡, π‘¦βŸ©}))
4037, 38, 39sylancl 587 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) ∧ ran 𝑓 = {𝑦}) β†’ (𝑓:{𝐡}⟢{𝑦} ↔ 𝑓 = {⟨𝐡, π‘¦βŸ©}))
4136, 40mpbid 231 . . . . . . . . . 10 (((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) ∧ ran 𝑓 = {𝑦}) β†’ 𝑓 = {⟨𝐡, π‘¦βŸ©})
4231, 41jca 513 . . . . . . . . 9 (((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) ∧ ran 𝑓 = {𝑦}) β†’ (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©}))
4342ex 414 . . . . . . . 8 ((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) β†’ (ran 𝑓 = {𝑦} β†’ (𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©})))
4443eximdv 1921 . . . . . . 7 ((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) β†’ (βˆƒπ‘¦ran 𝑓 = {𝑦} β†’ βˆƒπ‘¦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©})))
4524, 44mpd 15 . . . . . 6 ((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) β†’ βˆƒπ‘¦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©}))
46 df-rex 3071 . . . . . 6 (βˆƒπ‘¦ ∈ 𝐴 𝑓 = {⟨𝐡, π‘¦βŸ©} ↔ βˆƒπ‘¦(𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©}))
4745, 46sylibr 233 . . . . 5 ((πœ‘ ∧ 𝑓:{𝐡}⟢𝐴) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑓 = {⟨𝐡, π‘¦βŸ©})
4847ex 414 . . . 4 (πœ‘ β†’ (𝑓:{𝐡}⟢𝐴 β†’ βˆƒπ‘¦ ∈ 𝐴 𝑓 = {⟨𝐡, π‘¦βŸ©}))
49 f1osng 6826 . . . . . . . . . . 11 ((𝐡 ∈ π‘Š ∧ 𝑦 ∈ V) β†’ {⟨𝐡, π‘¦βŸ©}:{𝐡}–1-1-ontoβ†’{𝑦})
506, 38, 49sylancl 587 . . . . . . . . . 10 (πœ‘ β†’ {⟨𝐡, π‘¦βŸ©}:{𝐡}–1-1-ontoβ†’{𝑦})
5150adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©}) β†’ {⟨𝐡, π‘¦βŸ©}:{𝐡}–1-1-ontoβ†’{𝑦})
52 f1oeq1 6773 . . . . . . . . . . 11 (𝑓 = {⟨𝐡, π‘¦βŸ©} β†’ (𝑓:{𝐡}–1-1-ontoβ†’{𝑦} ↔ {⟨𝐡, π‘¦βŸ©}:{𝐡}–1-1-ontoβ†’{𝑦}))
5352bicomd 222 . . . . . . . . . 10 (𝑓 = {⟨𝐡, π‘¦βŸ©} β†’ ({⟨𝐡, π‘¦βŸ©}:{𝐡}–1-1-ontoβ†’{𝑦} ↔ 𝑓:{𝐡}–1-1-ontoβ†’{𝑦}))
5453adantl 483 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©}) β†’ ({⟨𝐡, π‘¦βŸ©}:{𝐡}–1-1-ontoβ†’{𝑦} ↔ 𝑓:{𝐡}–1-1-ontoβ†’{𝑦}))
5551, 54mpbid 231 . . . . . . . 8 ((πœ‘ ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©}) β†’ 𝑓:{𝐡}–1-1-ontoβ†’{𝑦})
56 f1of 6785 . . . . . . . 8 (𝑓:{𝐡}–1-1-ontoβ†’{𝑦} β†’ 𝑓:{𝐡}⟢{𝑦})
5755, 56syl 17 . . . . . . 7 ((πœ‘ ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©}) β†’ 𝑓:{𝐡}⟢{𝑦})
58573adant2 1132 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©}) β†’ 𝑓:{𝐡}⟢{𝑦})
59 snssi 4769 . . . . . . 7 (𝑦 ∈ 𝐴 β†’ {𝑦} βŠ† 𝐴)
60593ad2ant2 1135 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©}) β†’ {𝑦} βŠ† 𝐴)
6158, 60fssd 6687 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ 𝐴 ∧ 𝑓 = {⟨𝐡, π‘¦βŸ©}) β†’ 𝑓:{𝐡}⟢𝐴)
6261rexlimdv3a 3153 . . . 4 (πœ‘ β†’ (βˆƒπ‘¦ ∈ 𝐴 𝑓 = {⟨𝐡, π‘¦βŸ©} β†’ 𝑓:{𝐡}⟢𝐴))
6348, 62impbid 211 . . 3 (πœ‘ β†’ (𝑓:{𝐡}⟢𝐴 ↔ βˆƒπ‘¦ ∈ 𝐴 𝑓 = {⟨𝐡, π‘¦βŸ©}))
644, 63bitrd 279 . 2 (πœ‘ β†’ (𝑓 ∈ (𝐴 ↑m {𝐡}) ↔ βˆƒπ‘¦ ∈ 𝐴 𝑓 = {⟨𝐡, π‘¦βŸ©}))
6564abbi2dv 2868 1 (πœ‘ β†’ (𝐴 ↑m {𝐡}) = {𝑓 ∣ βˆƒπ‘¦ ∈ 𝐴 𝑓 = {⟨𝐡, π‘¦βŸ©}})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆƒ!weu 2563  {cab 2710  βˆƒwrex 3070  Vcvv 3444   βŠ† wss 3911  {csn 4587  βŸ¨cop 4593   class class class wbr 5106  dom cdm 5634  ran crn 5635   β€œ cima 5637  Rel wrel 5639   Fn wfn 6492  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  (class class class)co 7358   ↑m cmap 8768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770
This theorem is referenced by:  mapsn  8829  mapsnend  8983  iunmapsn  43525
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