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Theorem mapsnf1o3 8445
 Description: Explicit bijection in the reverse of mapsnf1o2 8444. (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 2798 . . . 4 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)) = (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋))
51, 2, 3, 4mapsnf1o2 8444 . . 3 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)):(𝐵m 𝑆)–1-1-onto𝐵
6 f1ocnv 6603 . . 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 8443 . . . 4 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)) = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
108, 9eqtr4i 2824 . . 3 𝐹 = (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋))
11 f1oeq1 6580 . . 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 234 1 𝐹:𝐵1-1-onto→(𝐵m 𝑆)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525   ↦ cmpt 5111   × cxp 5518  ◡ccnv 5519  –1-1-onto→wf1o 6324  ‘cfv 6325  (class class class)co 7136   ↑m cmap 8392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-map 8394 This theorem is referenced by:  coe1f2  20848  coe1add  20903  evls1rhmlem  20955  evl1sca  20968  pf1ind  20989  ismrer1  35295
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