Theorem dffun5r 4857

Description: A way of proving a relation is a function, analogous to mo2r 1949. (Contributed by Jim Kingdon, 27-May-2020.)

Assertion

Ref	Expression
dffun5r	⊢ ((Rel A ∧ ∀x∃z∀y(⟨x, y⟩ ∈ A → y = z)) → Fun A)

Distinct variable group:   x,y,z,A

Proof of Theorem dffun5r

Step	Hyp	Ref	Expression
1		nfv 1418	. . . . . 6 ⊢ Ⅎz⟨x, y⟩ ∈ A
2	1	mo2r 1949	. . . . 5 ⊢ (∃z∀y(⟨x, y⟩ ∈ A → y = z) → ∃*y⟨x, y⟩ ∈ A)
3		opeq2 3541	. . . . . . 7 ⊢ (y = z → ⟨x, y⟩ = ⟨x, z⟩)
4	3	eleq1d 2103	. . . . . 6 ⊢ (y = z → (⟨x, y⟩ ∈ A ↔ ⟨x, z⟩ ∈ A))
5	4	mo4 1958	. . . . 5 ⊢ (∃*y⟨x, y⟩ ∈ A ↔ ∀y∀z((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z))
6	2, 5	sylib 127	. . . 4 ⊢ (∃z∀y(⟨x, y⟩ ∈ A → y = z) → ∀y∀z((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z))
7	6	alimi 1341	. . 3 ⊢ (∀x∃z∀y(⟨x, y⟩ ∈ A → y = z) → ∀x∀y∀z((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z))
8	7	anim2i 324	. 2 ⊢ ((Rel A ∧ ∀x∃z∀y(⟨x, y⟩ ∈ A → y = z)) → (Rel A ∧ ∀x∀y∀z((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z)))
9		dffun4 4856	. 2 ⊢ (Fun A ↔ (Rel A ∧ ∀x∀y∀z((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z)))
10	8, 9	sylibr 137	1 ⊢ ((Rel A ∧ ∀x∃z∀y(⟨x, y⟩ ∈ A → y = z)) → Fun A)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240  ∃wex 1378   ∈ wcel 1390  ∃*wmo 1898  ⟨cop 3370  Rel wrel 4293  Fun wfun 4839

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-cnv 4296  df-co 4297  df-fun 4847

This theorem is referenced by: (None)