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Theorem funressnvmo 44539
Description: A function restricted to a singleton has at most one value for the singleton element as argument. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
funressnvmo (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
Distinct variable groups:   𝑦,𝐹   𝑥,𝑦
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem funressnvmo
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dffun6 6449 . 2 (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦))
2 breq1 5077 . . . . . . 7 (𝑥 = 𝑧 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
32equcoms 2023 . . . . . 6 (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
43biimpd 228 . . . . 5 (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
54moimdv 2546 . . . 4 (𝑧 = 𝑥 → (∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦))
65spimvw 1999 . . 3 (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦)
7 vsnid 4598 . . . . . 6 𝑥 ∈ {𝑥}
8 vex 3436 . . . . . . 7 𝑦 ∈ V
98brresi 5900 . . . . . 6 (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ (𝑥 ∈ {𝑥} ∧ 𝑥𝐹𝑦))
107, 9mpbiran 706 . . . . 5 (𝑥(𝐹 ↾ {𝑥})𝑦𝑥𝐹𝑦)
1110biimpri 227 . . . 4 (𝑥𝐹𝑦𝑥(𝐹 ↾ {𝑥})𝑦)
1211moimi 2545 . . 3 (∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦)
136, 12syl 17 . 2 (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦)
141, 13simplbiim 505 1 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537  wcel 2106  ∃*wmo 2538  {csn 4561   class class class wbr 5074  cres 5591  Rel wrel 5594  Fun wfun 6427
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-res 5601  df-fun 6435
This theorem is referenced by:  funressnmo  44540  funressndmafv2rn  44715
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