Theorem dffun4 2676

Description: Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24.

Assertion

Ref	Expression
dffun4	⊢ (Fun A ↔ (Rel A ∧ ∀x∀y∀z((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z)))

Distinct variable group(s):   x,y,z,A

Proof of Theorem dffun4

Step	Hyp	Ref	Expression
1		dffun2 2674	. 2 ⊢ (Fun A ↔ (Rel A ∧ ∀x∀y∀z((xAy ∧ xAz) → y = z)))
2		df-br 2063	. . . . . . 7 ⊢ (xAy ↔ ⟨x, y⟩ ∈ A)
3		df-br 2063	. . . . . . 7 ⊢ (xAz ↔ ⟨x, z⟩ ∈ A)
4	2, 3	anbi12i 369	. . . . . 6 ⊢ ((xAy ∧ xAz) ↔ (⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A))
5	4	imbi1i 161	. . . . 5 ⊢ (((xAy ∧ xAz) → y = z) ↔ ((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z))
6	5	bial 695	. . . 4 ⊢ (∀z((xAy ∧ xAz) → y = z) ↔ ∀z((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z))
7	6	bi2al 696	. . 3 ⊢ (∀x∀y∀z((xAy ∧ xAz) → y = z) ↔ ∀x∀y∀z((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z))
8	7	anbi2i 367	. 2 ⊢ ((Rel A ∧ ∀x∀y∀z((xAy ∧ xAz) → y = z)) ↔ (Rel A ∧ ∀x∀y∀z((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z)))
9	1, 8	bitr 151	1 ⊢ (Fun A ↔ (Rel A ∧ ∀x∀y∀z((⟨x, y⟩ ∈ A ∧ ⟨x, z⟩ ∈ A) → y = z)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  funsn 2690  funun 2700  fununi 2705  tfrlem7 2955

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-cnv 2426  df-co 2427  df-fun 2432

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