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Theorem fsetsnf1o 47003
Description: The mapping of an element of a class to a singleton function is a bijection. (Contributed by AV, 13-Sep-2024.)
Hypotheses
Ref Expression
fsetsnf.a 𝐴 = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
fsetsnf.f 𝐹 = (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩})
Assertion
Ref Expression
fsetsnf1o (𝑆𝑉𝐹:𝐵1-1-onto𝐴)
Distinct variable groups:   𝑥,𝐴   𝐵,𝑏,𝑥,𝑦   𝑆,𝑏,𝑥,𝑦   𝑉,𝑏,𝑥
Allowed substitution hints:   𝐴(𝑦,𝑏)   𝐹(𝑥,𝑦,𝑏)   𝑉(𝑦)

Proof of Theorem fsetsnf1o
StepHypRef Expression
1 fsetsnf.a . . 3 𝐴 = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
2 fsetsnf.f . . 3 𝐹 = (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩})
31, 2fsetsnf1 47001 . 2 (𝑆𝑉𝐹:𝐵1-1𝐴)
41, 2fsetsnfo 47002 . 2 (𝑆𝑉𝐹:𝐵onto𝐴)
5 df-f1o 6569 . 2 (𝐹:𝐵1-1-onto𝐴 ↔ (𝐹:𝐵1-1𝐴𝐹:𝐵onto𝐴))
63, 4, 5sylanbrc 583 1 (𝑆𝑉𝐹:𝐵1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  {cab 2711  wrex 3067  {csn 4630  cop 4636  cmpt 5230  1-1wf1 6559  ontowfo 6560  1-1-ontowf1o 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570
This theorem is referenced by:  fsetsnprcnex  47004
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