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Theorem funressnvmo 44101
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 6355 . 2 (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦))
2 breq1 5034 . . . . . . 7 (𝑥 = 𝑧 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
32equcoms 2032 . . . . . 6 (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
43biimpd 232 . . . . 5 (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
54moimdv 2547 . . . 4 (𝑧 = 𝑥 → (∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦))
65spimvw 2007 . . 3 (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦)
7 vsnid 4554 . . . . . 6 𝑥 ∈ {𝑥}
8 vex 3403 . . . . . . 7 𝑦 ∈ V
98brresi 5835 . . . . . 6 (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ (𝑥 ∈ {𝑥} ∧ 𝑥𝐹𝑦))
107, 9mpbiran 709 . . . . 5 (𝑥(𝐹 ↾ {𝑥})𝑦𝑥𝐹𝑦)
1110biimpri 231 . . . 4 (𝑥𝐹𝑦𝑥(𝐹 ↾ {𝑥})𝑦)
1211moimi 2546 . . 3 (∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦)
136, 12syl 17 . 2 (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦)
141, 13simplbiim 508 1 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1540  wcel 2114  ∃*wmo 2539  {csn 4517   class class class wbr 5031  cres 5528  Rel wrel 5531  Fun wfun 6334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-br 5032  df-opab 5094  df-id 5430  df-xp 5532  df-cnv 5534  df-co 5535  df-res 5538  df-fun 6342
This theorem is referenced by:  funressnmo  44102  funressndmafv2rn  44278
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