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Theorem mapsnf1o2 8073
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 6363 . . 3 (𝑥𝑋) ∈ V
2 mapsncnv.f . . 3 𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
31, 2fnmpti 6183 . 2 𝐹 Fn (𝐵𝑚 𝑆)
4 mapsncnv.s . . . . 5 𝑆 = {𝑋}
5 snex 5057 . . . . 5 {𝑋} ∈ V
64, 5eqeltri 2835 . . . 4 𝑆 ∈ V
7 snex 5057 . . . 4 {𝑦} ∈ V
86, 7xpex 7128 . . 3 (𝑆 × {𝑦}) ∈ V
9 mapsncnv.b . . . 4 𝐵 ∈ V
10 mapsncnv.x . . . 4 𝑋 ∈ V
114, 9, 10, 2mapsncnv 8072 . . 3 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
128, 11fnmpti 6183 . 2 𝐹 Fn 𝐵
13 dff1o4 6307 . 2 (𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵 ↔ (𝐹 Fn (𝐵𝑚 𝑆) ∧ 𝐹 Fn 𝐵))
143, 12, 13mpbir2an 993 1 𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321  cmpt 4881   × cxp 5264  ccnv 5265   Fn wfn 6044  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6814  𝑚 cmap 8025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027
This theorem is referenced by:  mapsnf1o3  8074  coe1sfi  19805  coe1mul2lem2  19860  ply1coe  19888  evl1var  19922  pf1mpf  19938  pf1ind  19941  deg1ldg  24071  deg1leb  24074
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