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Theorem fsetsnf1 45762
Description: The mapping of an element of a class to a singleton function is an injection. (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
fsetsnf1 (𝑆𝑉𝐹:𝐵1-1𝐴)
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 fsetsnf1
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 45761 . 2 (𝑆𝑉𝐹:𝐵𝐴)
42a1i 11 . . . . . . 7 ((𝑚𝐵𝑛𝐵) → 𝐹 = (𝑥𝐵 ↦ {⟨𝑆, 𝑥⟩}))
5 opeq2 4875 . . . . . . . . 9 (𝑥 = 𝑚 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑚⟩)
65sneqd 4641 . . . . . . . 8 (𝑥 = 𝑚 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑚⟩})
76adantl 483 . . . . . . 7 (((𝑚𝐵𝑛𝐵) ∧ 𝑥 = 𝑚) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑚⟩})
8 simpl 484 . . . . . . 7 ((𝑚𝐵𝑛𝐵) → 𝑚𝐵)
9 snex 5432 . . . . . . . 8 {⟨𝑆, 𝑚⟩} ∈ V
109a1i 11 . . . . . . 7 ((𝑚𝐵𝑛𝐵) → {⟨𝑆, 𝑚⟩} ∈ V)
114, 7, 8, 10fvmptd 7006 . . . . . 6 ((𝑚𝐵𝑛𝐵) → (𝐹𝑚) = {⟨𝑆, 𝑚⟩})
12 opeq2 4875 . . . . . . . . 9 (𝑥 = 𝑛 → ⟨𝑆, 𝑥⟩ = ⟨𝑆, 𝑛⟩)
1312sneqd 4641 . . . . . . . 8 (𝑥 = 𝑛 → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
1413adantl 483 . . . . . . 7 (((𝑚𝐵𝑛𝐵) ∧ 𝑥 = 𝑛) → {⟨𝑆, 𝑥⟩} = {⟨𝑆, 𝑛⟩})
15 simpr 486 . . . . . . 7 ((𝑚𝐵𝑛𝐵) → 𝑛𝐵)
16 snex 5432 . . . . . . . 8 {⟨𝑆, 𝑛⟩} ∈ V
1716a1i 11 . . . . . . 7 ((𝑚𝐵𝑛𝐵) → {⟨𝑆, 𝑛⟩} ∈ V)
184, 14, 15, 17fvmptd 7006 . . . . . 6 ((𝑚𝐵𝑛𝐵) → (𝐹𝑛) = {⟨𝑆, 𝑛⟩})
1911, 18eqeq12d 2749 . . . . 5 ((𝑚𝐵𝑛𝐵) → ((𝐹𝑚) = (𝐹𝑛) ↔ {⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩}))
2019adantl 483 . . . 4 ((𝑆𝑉 ∧ (𝑚𝐵𝑛𝐵)) → ((𝐹𝑚) = (𝐹𝑛) ↔ {⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩}))
21 opex 5465 . . . . . 6 𝑆, 𝑚⟩ ∈ V
2221sneqr 4842 . . . . 5 ({⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩} → ⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩)
23 opthg 5478 . . . . . . 7 ((𝑆𝑉𝑚𝐵) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ ↔ (𝑆 = 𝑆𝑚 = 𝑛)))
2423adantrr 716 . . . . . 6 ((𝑆𝑉 ∧ (𝑚𝐵𝑛𝐵)) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ ↔ (𝑆 = 𝑆𝑚 = 𝑛)))
25 simpr 486 . . . . . 6 ((𝑆 = 𝑆𝑚 = 𝑛) → 𝑚 = 𝑛)
2624, 25syl6bi 253 . . . . 5 ((𝑆𝑉 ∧ (𝑚𝐵𝑛𝐵)) → (⟨𝑆, 𝑚⟩ = ⟨𝑆, 𝑛⟩ → 𝑚 = 𝑛))
2722, 26syl5 34 . . . 4 ((𝑆𝑉 ∧ (𝑚𝐵𝑛𝐵)) → ({⟨𝑆, 𝑚⟩} = {⟨𝑆, 𝑛⟩} → 𝑚 = 𝑛))
2820, 27sylbid 239 . . 3 ((𝑆𝑉 ∧ (𝑚𝐵𝑛𝐵)) → ((𝐹𝑚) = (𝐹𝑛) → 𝑚 = 𝑛))
2928ralrimivva 3201 . 2 (𝑆𝑉 → ∀𝑚𝐵𝑛𝐵 ((𝐹𝑚) = (𝐹𝑛) → 𝑚 = 𝑛))
30 dff13 7254 . 2 (𝐹:𝐵1-1𝐴 ↔ (𝐹:𝐵𝐴 ∧ ∀𝑚𝐵𝑛𝐵 ((𝐹𝑚) = (𝐹𝑛) → 𝑚 = 𝑛)))
313, 29, 30sylanbrc 584 1 (𝑆𝑉𝐹:𝐵1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wrex 3071  Vcvv 3475  {csn 4629  cop 4635  cmpt 5232  wf 6540  1-1wf1 6541  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fv 6552
This theorem is referenced by:  fsetsnf1o  45764
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