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Theorem mapsnf1o3 8935
Description: Explicit bijection in the reverse of mapsnf1o2 8934. (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 2737 . . . 4 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)) = (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋))
51, 2, 3, 4mapsnf1o2 8934 . . 3 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)):(𝐵m 𝑆)–1-1-onto𝐵
6 f1ocnv 6860 . . 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 8933 . . . 4 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)) = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
108, 9eqtr4i 2768 . . 3 𝐹 = (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋))
11 f1oeq1 6836 . . 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 231 1 𝐹:𝐵1-1-onto→(𝐵m 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cmpt 5225   × cxp 5683  ccnv 5684  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by:  coe1f2  22211  coe1add  22267  evls1rhmlem  22325  evl1sca  22338  pf1ind  22359  ismrer1  37845
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