Theorem ndmoprg 3057

Description: The value of an operation outside its domain.

Hypothesis

Ref	Expression
ndmoprg.1	⊢ dom F = (S × S)

Assertion

Ref	Expression
ndmoprg	⊢ ((B ∈ C ∧ ¬ (A ∈ S ∧ B ∈ S)) → (AFB) = ∅)

Proof of Theorem ndmoprg

Step	Hyp	Ref	Expression
1		opelxpg 2454	. . . . 5 ⊢ (B ∈ C → (⟨A, B⟩ ∈ (S × S) ↔ (A ∈ S ∧ B ∈ S)))
2		ndmoprg.1	. . . . . 6 ⊢ dom F = (S × S)
3	2	eleq2i 1153	. . . . 5 ⊢ (⟨A, B⟩ ∈ dom F ↔ ⟨A, B⟩ ∈ (S × S))
4	1, 3	syl5bb 410	. . . 4 ⊢ (B ∈ C → (⟨A, B⟩ ∈ dom F ↔ (A ∈ S ∧ B ∈ S)))
5	4	negbid 463	. . 3 ⊢ (B ∈ C → (¬ ⟨A, B⟩ ∈ dom F ↔ ¬ (A ∈ S ∧ B ∈ S)))
6		ndmfv 2848	. . . 4 ⊢ (¬ ⟨A, B⟩ ∈ dom F → (F ‘⟨A, B⟩) = ∅)
7		df-opr 3003	. . . 4 ⊢ (AFB) = (F ‘⟨A, B⟩)
8	6, 7	syl5eq 1136	. . 3 ⊢ (¬ ⟨A, B⟩ ∈ dom F → (AFB) = ∅)
9	5, 8	syl6bir 188	. 2 ⊢ (B ∈ C → (¬ (A ∈ S ∧ B ∈ S) → (AFB) = ∅))
10	9	imp 277	1 ⊢ ((B ∈ C ∧ ¬ (A ∈ S ∧ B ∈ S)) → (AFB) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∅c0 1707  ⟨cop 1810   × cxp 2408  dom cdm 2410   ‘cfv 2422  (class class class)co 3001

This theorem is referenced by:  ndmoprcl 3058  ndmopr 3059

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-eu 1009  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  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-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

metamath.org