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Theorem mapsnf1o2 7849
Description: Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypotheses
Ref Expression
mapsncnv.s 𝑆 = {𝑋}
mapsncnv.b 𝐵 ∈ V
mapsncnv.x 𝑋 ∈ V
mapsncnv.f 𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
Assertion
Ref Expression
mapsnf1o2 𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵
Distinct variable groups:   𝑥,𝐵   𝑥,𝑆
Allowed substitution hints:   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem mapsnf1o2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvex 6158 . . 3 (𝑥𝑋) ∈ V
2 mapsncnv.f . . 3 𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
31, 2fnmpti 5979 . 2 𝐹 Fn (𝐵𝑚 𝑆)
4 mapsncnv.s . . . . 5 𝑆 = {𝑋}
5 snex 4869 . . . . 5 {𝑋} ∈ V
64, 5eqeltri 2694 . . . 4 𝑆 ∈ V
7 snex 4869 . . . 4 {𝑦} ∈ V
86, 7xpex 6915 . . 3 (𝑆 × {𝑦}) ∈ V
9 mapsncnv.b . . . 4 𝐵 ∈ V
10 mapsncnv.x . . . 4 𝑋 ∈ V
114, 9, 10, 2mapsncnv 7848 . . 3 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
128, 11fnmpti 5979 . 2 𝐹 Fn 𝐵
13 dff1o4 6102 . 2 (𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵 ↔ (𝐹 Fn (𝐵𝑚 𝑆) ∧ 𝐹 Fn 𝐵))
143, 12, 13mpbir2an 954 1 𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4148  cmpt 4673   × cxp 5072  ccnv 5073   Fn wfn 5842  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by:  mapsnf1o3  7850  coe1sfi  19502  coe1mul2lem2  19557  ply1coe  19585  evl1var  19619  pf1mpf  19635  pf1ind  19638  deg1ldg  23756  deg1leb  23759
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