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Theorem funressnvmo 43270
 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 6363 . 2 (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦))
2 breq1 5060 . . . . . . 7 (𝑥 = 𝑧 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
32equcoms 2020 . . . . . 6 (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
43biimpd 231 . . . . 5 (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
54moimdv 2623 . . . 4 (𝑧 = 𝑥 → (∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦))
65spimvw 1995 . . 3 (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦)
7 vsnid 4594 . . . . . 6 𝑥 ∈ {𝑥}
8 vex 3496 . . . . . . 7 𝑦 ∈ V
98brresi 5855 . . . . . 6 (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ (𝑥 ∈ {𝑥} ∧ 𝑥𝐹𝑦))
107, 9mpbiran 707 . . . . 5 (𝑥(𝐹 ↾ {𝑥})𝑦𝑥𝐹𝑦)
1110biimpri 230 . . . 4 (𝑥𝐹𝑦𝑥(𝐹 ↾ {𝑥})𝑦)
1211moimi 2621 . . 3 (∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦)
136, 12syl 17 . 2 (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦)
141, 13simplbiim 507 1 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1528   ∈ wcel 2107  ∃*wmo 2614  {csn 4559   class class class wbr 5057   ↾ cres 5550  Rel wrel 5553  Fun wfun 6342 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-cnv 5556  df-co 5557  df-res 5560  df-fun 6350 This theorem is referenced by:  funressnmo  43271  funressndmafv2rn  43412
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