Theorem ndmima 2623

Description: The image of a singleton outside the domain is empty.

Assertion

Ref	Expression
ndmima	⊢ (¬ A ∈ dom B → (B “ {A}) = ∅)

Proof of Theorem ndmima

Step	Hyp	Ref	Expression
1		disjsn 1836	. . . . 5 ⊢ ((dom B ∩ {A}) = ∅ ↔ ¬ A ∈ dom B)
2	1	biimpr 134	. . . 4 ⊢ (¬ A ∈ dom B → (dom B ∩ {A}) = ∅)
3		dmres 2584	. . . . 5 ⊢ dom (B ↾ {A}) = ({A} ∩ dom B)
4		incom 1636	. . . . 5 ⊢ ({A} ∩ dom B) = (dom B ∩ {A})
5	3, 4	eqtr 1119	. . . 4 ⊢ dom (B ↾ {A}) = (dom B ∩ {A})
6	2, 5	syl5eq 1136	. . 3 ⊢ (¬ A ∈ dom B → dom (B ↾ {A}) = ∅)
7		dm0rn0 2549	. . 3 ⊢ (dom (B ↾ {A}) = ∅ ↔ ran (B ↾ {A}) = ∅)
8	6, 7	sylib 173	. 2 ⊢ (¬ A ∈ dom B → ran (B ↾ {A}) = ∅)
9		df-ima 2431	. 2 ⊢ (B “ {A}) = ran (B ↾ {A})
10	8, 9	syl5eq 1136	1 ⊢ (¬ A ∈ dom B → (B “ {A}) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092   ∩ cin 1486  ∅c0 1707  {csn 1808  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413

This theorem is referenced by:  funfv 2862

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-ral 1205  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-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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