Theorem funmo 6497

Description: A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) (Proof shortened by SN, 19-Dec-2024.)

Assertion

Ref	Expression
funmo	⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem funmo

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun6 6492	. . . . . 6 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
2	1	simplbi 497	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
3		brrelex1 5669	. . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑦) → 𝐴 ∈ V)
4	3	ex 412	. . . . 5 ⊢ (Rel 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V))
5	2, 4	syl 17	. . . 4 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V))
6	5	ancrd 551	. . 3 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
7	6	alrimiv 1928	. 2 ⊢ (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
8	1	simprbi 496	. . . 4 ⊢ (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
9		breq1 5094	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
10	9	mobidv 2544	. . . . 5 ⊢ (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
11	10	spcgv 3551	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥∃*𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝐴𝐹𝑦))
12	8, 11	syl5com 31	. . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
13		moanimv 2614	. . 3 ⊢ (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
14	12, 13	sylibr 234	. 2 ⊢ (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦))
15		moim 2539	. 2 ⊢ (∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)) → (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) → ∃*𝑦 𝐴𝐹𝑦))
16	7, 14, 15	sylc 65	1 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2111  ∃*wmo 2533  Vcvv 3436   class class class wbr 5091  Rel wrel 5621  Fun wfun 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-fun 6483

This theorem is referenced by:  funeu  6506  funco  6521  fununmo  6528  imadif  6565  fneu  6591  dff3  7033  shftfn  14980