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Theorem mapsnf1o3 8064
Description: Explicit bijection in the reverse of mapsnf1o2 8063. (Contributed by Stefan O'Rear, 24-Mar-2015.)
Hypotheses
Ref Expression
mapsncnv.s 𝑆 = {𝑋}
mapsncnv.b 𝐵 ∈ V
mapsncnv.x 𝑋 ∈ V
mapsnf1o3.f 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
Assertion
Ref Expression
mapsnf1o3 𝐹:𝐵1-1-onto→(𝐵𝑚 𝑆)
Distinct variable groups:   𝑦,𝐵   𝑦,𝑆   𝑦,𝑋
Allowed substitution hint:   𝐹(𝑦)

Proof of Theorem mapsnf1o3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mapsncnv.s . . . 4 𝑆 = {𝑋}
2 mapsncnv.b . . . 4 𝐵 ∈ V
3 mapsncnv.x . . . 4 𝑋 ∈ V
4 eqid 2771 . . . 4 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)) = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
51, 2, 3, 4mapsnf1o2 8063 . . 3 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)):(𝐵𝑚 𝑆)–1-1-onto𝐵
6 f1ocnv 6291 . . 3 ((𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)):(𝐵𝑚 𝑆)–1-1-onto𝐵(𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)):𝐵1-1-onto→(𝐵𝑚 𝑆))
75, 6ax-mp 5 . 2 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)):𝐵1-1-onto→(𝐵𝑚 𝑆)
8 mapsnf1o3.f . . . 4 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
91, 2, 3, 4mapsncnv 8062 . . . 4 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)) = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
108, 9eqtr4i 2796 . . 3 𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
11 f1oeq1 6269 . . 3 (𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)) → (𝐹:𝐵1-1-onto→(𝐵𝑚 𝑆) ↔ (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)):𝐵1-1-onto→(𝐵𝑚 𝑆)))
1210, 11ax-mp 5 . 2 (𝐹:𝐵1-1-onto→(𝐵𝑚 𝑆) ↔ (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)):𝐵1-1-onto→(𝐵𝑚 𝑆))
137, 12mpbir 221 1 𝐹:𝐵1-1-onto→(𝐵𝑚 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4317  cmpt 4864   × cxp 5248  ccnv 5249  1-1-ontowf1o 6029  cfv 6030  (class class class)co 6796  𝑚 cmap 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-map 8015
This theorem is referenced by:  coe1f2  19794  coe1add  19849  evls1rhmlem  19901  evl1sca  19913  pf1ind  19934  ismrer1  33969
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