Theorem funmo 5372

Description: A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)

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 5371	. . . . . 6 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
2	1	simplbi 274	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
3		brrelex 4795	. . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑦) → 𝐴 ∈ V)
4	3	ex 115	. . . . 5 ⊢ (Rel 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V))
5	2, 4	syl 14	. . . 4 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → 𝐴 ∈ V))
6	5	ancrd 326	. . 3 ⊢ (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
7	6	alrimiv 1923	. 2 ⊢ (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
8		breq1 4117	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
9	8	mobidv 2118	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
10	9	imbi2d 230	. . . . 5 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)))
11	1	simprbi 275	. . . . . 6 ⊢ (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
12	11	19.21bi 1607	. . . . 5 ⊢ (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
13	10, 12	vtoclg 2877	. . . 4 ⊢ (𝐴 ∈ V → (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦))
14	13	com12 30	. . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
15		moanimv 2158	. . 3 ⊢ (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
16	14, 15	sylibr 134	. 2 ⊢ (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦))
17		moim 2147	. 2 ⊢ (∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)) → (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) → ∃*𝑦 𝐴𝐹𝑦))
18	7, 16, 17	sylc 62	1 ⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396   = wceq 1398  ∃*wmo 2083   ∈ wcel 2205  Vcvv 2815   class class class wbr 4114  Rel wrel 4759  Fun wfun 5351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-fun 5359

This theorem is referenced by:  funeu  5382  funco  5397  fununmo  5403  imadif  5441  fneu  5467  dff3im  5827  shftfn  11534