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Theorem mapsn 6026
 Description: The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by set.mm contributors, 10-Dec-2003.)
Hypotheses
Ref Expression
map0.1 A V
map0.2 B V
Assertion
Ref Expression
mapsn (Am {B}) = {f y A f = {B, y}}
Distinct variable groups:   y,f,A   B,f,y

Proof of Theorem mapsn
StepHypRef Expression
1 map0.1 . . 3 A V
2 snex 4111 . . 3 {B} V
31, 2mapval 6011 . 2 (Am {B}) = {f f:{B}–→A}
4 ffn 5223 . . . . . . . 8 (f:{B}–→Af Fn {B})
5 map0.2 . . . . . . . . 9 B V
65snid 3760 . . . . . . . 8 B {B}
7 fneu 5187 . . . . . . . 8 ((f Fn {B} B {B}) → ∃!y Bfy)
84, 6, 7sylancl 643 . . . . . . 7 (f:{B}–→A∃!y Bfy)
9 euabsn 3792 . . . . . . . 8 (∃!y Bfyy{y Bfy} = {y})
10 imadmrn 5008 . . . . . . . . . . . 12 (f “ dom f) = ran f
11 fdm 5226 . . . . . . . . . . . . 13 (f:{B}–→A → dom f = {B})
1211imaeq2d 4942 . . . . . . . . . . . 12 (f:{B}–→A → (f “ dom f) = (f “ {B}))
1310, 12syl5eqr 2399 . . . . . . . . . . 11 (f:{B}–→A → ran f = (f “ {B}))
14 imasn 5018 . . . . . . . . . . 11 (f “ {B}) = {y Bfy}
1513, 14syl6req 2402 . . . . . . . . . 10 (f:{B}–→A → {y Bfy} = ran f)
1615eqeq1d 2361 . . . . . . . . 9 (f:{B}–→A → ({y Bfy} = {y} ↔ ran f = {y}))
1716exbidv 1626 . . . . . . . 8 (f:{B}–→A → (y{y Bfy} = {y} ↔ yran f = {y}))
189, 17syl5bb 248 . . . . . . 7 (f:{B}–→A → (∃!y Bfyyran f = {y}))
198, 18mpbid 201 . . . . . 6 (f:{B}–→Ayran f = {y})
20 frn 5228 . . . . . . . . 9 (f:{B}–→A → ran f A)
21 sseq1 3292 . . . . . . . . . 10 (ran f = {y} → (ran f A ↔ {y} A))
22 vex 2862 . . . . . . . . . . 11 y V
2322snss 3838 . . . . . . . . . 10 (y A ↔ {y} A)
2421, 23syl6bbr 254 . . . . . . . . 9 (ran f = {y} → (ran f Ay A))
2520, 24syl5ibcom 211 . . . . . . . 8 (f:{B}–→A → (ran f = {y} → y A))
26 dffn4 5275 . . . . . . . . . . . 12 (f Fn {B} ↔ f:{B}–onto→ran f)
274, 26sylib 188 . . . . . . . . . . 11 (f:{B}–→Af:{B}–onto→ran f)
28 fof 5269 . . . . . . . . . . 11 (f:{B}–onto→ran ff:{B}–→ran f)
2927, 28syl 15 . . . . . . . . . 10 (f:{B}–→Af:{B}–→ran f)
30 feq3 5212 . . . . . . . . . 10 (ran f = {y} → (f:{B}–→ran ff:{B}–→{y}))
3129, 30syl5ibcom 211 . . . . . . . . 9 (f:{B}–→A → (ran f = {y} → f:{B}–→{y}))
325, 22fsn 5432 . . . . . . . . 9 (f:{B}–→{y} ↔ f = {B, y})
3331, 32syl6ib 217 . . . . . . . 8 (f:{B}–→A → (ran f = {y} → f = {B, y}))
3425, 33jcad 519 . . . . . . 7 (f:{B}–→A → (ran f = {y} → (y A f = {B, y})))
3534eximdv 1622 . . . . . 6 (f:{B}–→A → (yran f = {y} → y(y A f = {B, y})))
3619, 35mpd 14 . . . . 5 (f:{B}–→Ay(y A f = {B, y}))
37 df-rex 2620 . . . . 5 (y A f = {B, y} ↔ y(y A f = {B, y}))
3836, 37sylibr 203 . . . 4 (f:{B}–→Ay A f = {B, y})
395, 22f1osn 5322 . . . . . . . . 9 {B, y}:{B}–1-1-onto→{y}
40 f1of 5287 . . . . . . . . 9 ({B, y}:{B}–1-1-onto→{y} → {B, y}:{B}–→{y})
4139, 40ax-mp 8 . . . . . . . 8 {B, y}:{B}–→{y}
42 feq1 5210 . . . . . . . 8 (f = {B, y} → (f:{B}–→{y} ↔ {B, y}:{B}–→{y}))
4341, 42mpbiri 224 . . . . . . 7 (f = {B, y} → f:{B}–→{y})
44 snssi 3852 . . . . . . 7 (y A → {y} A)
45 fss 5230 . . . . . . 7 ((f:{B}–→{y} {y} A) → f:{B}–→A)
4643, 44, 45syl2an 463 . . . . . 6 ((f = {B, y} y A) → f:{B}–→A)
4746expcom 424 . . . . 5 (y A → (f = {B, y} → f:{B}–→A))
4847rexlimiv 2732 . . . 4 (y A f = {B, y} → f:{B}–→A)
4938, 48impbii 180 . . 3 (f:{B}–→Ay A f = {B, y})
5049abbii 2465 . 2 {f f:{B}–→A} = {f y A f = {B, y}}
513, 50eqtri 2373 1 (Am {B}) = {f y A f = {B, y}}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  {csn 3737  ⟨cop 4561   class class class wbr 4639   “ cima 4722  dom cdm 4772  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779  –1-1-onto→wf1o 4780  (class class class)co 5525   ↑m cmap 5999 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-map 6001 This theorem is referenced by: (None)
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