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Theorem fsetsnfo 47003
Description: The mapping of an element of a class to a singleton function is a surjection. (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
fsetsnfo (𝑆𝑉𝐹:𝐵onto𝐴)
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 fsetsnfo
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsetsnf.a . . 3 𝐴 = {𝑦 ∣ ∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}
2 fsetsnf.f . . 3 𝐹 = (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩})
31, 2fsetsnf 47001 . 2 (𝑆𝑉𝐹:𝐵𝐴)
4 vex 3482 . . . . . 6 𝑚 ∈ V
5 eqeq1 2739 . . . . . . 7 (𝑦 = 𝑚 → (𝑦 = {⟨𝑆, 𝑏⟩} ↔ 𝑚 = {⟨𝑆, 𝑏⟩}))
65rexbidv 3177 . . . . . 6 (𝑦 = 𝑚 → (∃𝑏𝐵 𝑦 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑏𝐵 𝑚 = {⟨𝑆, 𝑏⟩}))
74, 6, 1elab2 3685 . . . . 5 (𝑚𝐴 ↔ ∃𝑏𝐵 𝑚 = {⟨𝑆, 𝑏⟩})
8 opeq2 4879 . . . . . . . . 9 (𝑏 = 𝑛 → ⟨𝑆, 𝑏⟩ = ⟨𝑆, 𝑛⟩)
98sneqd 4643 . . . . . . . 8 (𝑏 = 𝑛 → {⟨𝑆, 𝑏⟩} = {⟨𝑆, 𝑛⟩})
109eqeq2d 2746 . . . . . . 7 (𝑏 = 𝑛 → (𝑚 = {⟨𝑆, 𝑏⟩} ↔ 𝑚 = {⟨𝑆, 𝑛⟩}))
1110cbvrexvw 3236 . . . . . 6 (∃𝑏𝐵 𝑚 = {⟨𝑆, 𝑏⟩} ↔ ∃𝑛𝐵 𝑚 = {⟨𝑆, 𝑛⟩})
12 simpr 484 . . . . . . . . 9 (((𝑆𝑉𝑛𝐵) ∧ 𝑚 = {⟨𝑆, 𝑛⟩}) → 𝑚 = {⟨𝑆, 𝑛⟩})
132a1i 11 . . . . . . . . . . . 12 ((𝑆𝑉𝑛𝐵) → 𝐹 = (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩}))
14 opeq2 4879 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑛⟩)
1514sneqd 4643 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
1615adantl 481 . . . . . . . . . . . 12 (((𝑆𝑉𝑛𝐵) ∧ 𝑥 = 𝑛) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
17 simpr 484 . . . . . . . . . . . 12 ((𝑆𝑉𝑛𝐵) → 𝑛𝐵)
18 snex 5442 . . . . . . . . . . . . 13 {⟨𝑆, 𝑛⟩} ∈ V
1918a1i 11 . . . . . . . . . . . 12 ((𝑆𝑉𝑛𝐵) → {⟨𝑆, 𝑛⟩} ∈ V)
2013, 16, 17, 19fvmptd 7023 . . . . . . . . . . 11 ((𝑆𝑉𝑛𝐵) → (𝐹𝑛) = {⟨𝑆, 𝑛⟩})
2120eqcomd 2741 . . . . . . . . . 10 ((𝑆𝑉𝑛𝐵) → {⟨𝑆, 𝑛⟩} = (𝐹𝑛))
2221adantr 480 . . . . . . . . 9 (((𝑆𝑉𝑛𝐵) ∧ 𝑚 = {⟨𝑆, 𝑛⟩}) → {⟨𝑆, 𝑛⟩} = (𝐹𝑛))
2312, 22eqtrd 2775 . . . . . . . 8 (((𝑆𝑉𝑛𝐵) ∧ 𝑚 = {⟨𝑆, 𝑛⟩}) → 𝑚 = (𝐹𝑛))
2423ex 412 . . . . . . 7 ((𝑆𝑉𝑛𝐵) → (𝑚 = {⟨𝑆, 𝑛⟩} → 𝑚 = (𝐹𝑛)))
2524reximdva 3166 . . . . . 6 (𝑆𝑉 → (∃𝑛𝐵 𝑚 = {⟨𝑆, 𝑛⟩} → ∃𝑛𝐵 𝑚 = (𝐹𝑛)))
2611, 25biimtrid 242 . . . . 5 (𝑆𝑉 → (∃𝑏𝐵 𝑚 = {⟨𝑆, 𝑏⟩} → ∃𝑛𝐵 𝑚 = (𝐹𝑛)))
277, 26biimtrid 242 . . . 4 (𝑆𝑉 → (𝑚𝐴 → ∃𝑛𝐵 𝑚 = (𝐹𝑛)))
2827imp 406 . . 3 ((𝑆𝑉𝑚𝐴) → ∃𝑛𝐵 𝑚 = (𝐹𝑛))
2928ralrimiva 3144 . 2 (𝑆𝑉 → ∀𝑚𝐴𝑛𝐵 𝑚 = (𝐹𝑛))
30 dffo3 7122 . 2 (𝐹:𝐵onto𝐴 ↔ (𝐹:𝐵𝐴 ∧ ∀𝑚𝐴𝑛𝐵 𝑚 = (𝐹𝑛)))
313, 29, 30sylanbrc 583 1 (𝑆𝑉𝐹:𝐵onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  {csn 4631  cop 4637  cmpt 5231  wf 6559  ontowfo 6561  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571
This theorem is referenced by:  fsetsnf1o  47004
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