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Theorem funressnvmo 43157
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 2018 . . . . . 6 (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
43biimpd 230 . . . . 5 (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
54moimdv 2622 . . . 4 (𝑧 = 𝑥 → (∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦))
65spimvw 1993 . . 3 (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦)
7 vsnid 4592 . . . . . 6 𝑥 ∈ {𝑥}
8 vex 3495 . . . . . . 7 𝑦 ∈ V
98brresi 5855 . . . . . 6 (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ (𝑥 ∈ {𝑥} ∧ 𝑥𝐹𝑦))
107, 9mpbiran 705 . . . . 5 (𝑥(𝐹 ↾ {𝑥})𝑦𝑥𝐹𝑦)
1110biimpri 229 . . . 4 (𝑥𝐹𝑦𝑥(𝐹 ↾ {𝑥})𝑦)
1211moimi 2620 . . 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 207  wal 1526  wcel 2105  ∃*wmo 2613  {csn 4557   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 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  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  43158  funressndmafv2rn  43299
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