Theorem funressnvmo 47706

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.

Step	Hyp	Ref	Expression
1		dffun6 6548	. 2 ⊢ (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦))
2		breq1 5114	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑧(𝐹 ↾ {𝑥})𝑦))
3	2	equcoms 2047	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑧(𝐹 ↾ {𝑥})𝑦))
4	3	biimpd 232	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦 → 𝑧(𝐹 ↾ {𝑥})𝑦))
5	4	moimdv 2580	. . . 4 ⊢ (𝑧 = 𝑥 → (∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦))
6	5	spimvw 2013	. . 3 ⊢ (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦)
7		vsnid 4632	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
8		vex 3465	. . . . . . 7 ⊢ 𝑦 ∈ V
9	8	brresi 5988	. . . . . 6 ⊢ (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ (𝑥 ∈ {𝑥} ∧ 𝑥𝐹𝑦))
10	7, 9	mpbiran 721	. . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑥𝐹𝑦)
11	10	biimpri 231	. . . 4 ⊢ (𝑥𝐹𝑦 → 𝑥(𝐹 ↾ {𝑥})𝑦)
12	11	moimi 2579	. . 3 ⊢ (∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦)
13	6, 12	syl 18	. 2 ⊢ (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦)
14	1, 13	simplbiim 513	1 ⊢ (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565   ∈ wcel 2149  ∃*wmo 2571  {csn 4592   class class class wbr 5111   ↾ cres 5664  Rel wrel 5667  Fun wfun 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5259  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-br 5112  df-opab 5176  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-res 5674  df-fun 6539

This theorem is referenced by:  funressnmo  47707  funressndmafv2rn  47884