Theorem funressnvmo 47638

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 6536	. 2 ⊢ (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦))
2		breq1 5107	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ 𝑧(𝐹 ↾ {𝑥})𝑦))
3	2	equcoms 2043	. . . . . 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 2576	. . . 4 ⊢ (𝑧 = 𝑥 → (∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦))
6	5	spimvw 2009	. . 3 ⊢ (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦)
7		vsnid 4625	. . . . . 6 ⊢ 𝑥 ∈ {𝑥}
8		vex 3461	. . . . . . 7 ⊢ 𝑦 ∈ V
9	8	brresi 5977	. . . . . 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 2575	. . 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 1561   ∈ wcel 2145  ∃*wmo 2567  {csn 4585   class class class wbr 5104   ↾ cres 5653  Rel wrel 5656  Fun wfun 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-res 5663  df-fun 6527

This theorem is referenced by:  funressnmo  47639  funressndmafv2rn  47816