Theorem ndmfvrcl 2849

Description: Reverse closure law for function with the empty set not in its domain.

Hypotheses

Ref	Expression
ndmfvrcl.1	⊢ dom F = S
ndmfvrcl.2	⊢ ¬ ∅ ∈ S

Assertion

Ref	Expression
ndmfvrcl	⊢ ((F ‘A) ∈ S → A ∈ S)

Proof of Theorem ndmfvrcl

Step	Hyp	Ref	Expression
1		ndmfvrcl.2	. . . 4 ⊢ ¬ ∅ ∈ S
2		ndmfv 2848	. . . . 5 ⊢ (¬ A ∈ dom F → (F ‘A) = ∅)
3	2	eleq1d 1155	. . . 4 ⊢ (¬ A ∈ dom F → ((F ‘A) ∈ S ↔ ∅ ∈ S))
4	1, 3	mtbiri 539	. . 3 ⊢ (¬ A ∈ dom F → ¬ (F ‘A) ∈ S)
5	4	a3i 69	. 2 ⊢ ((F ‘A) ∈ S → A ∈ dom F)
6		ndmfvrcl.1	. . 3 ⊢ dom F = S
7	6	eleq2i 1153	. 2 ⊢ (A ∈ dom F ↔ A ∈ S)
8	5, 7	sylib 173	1 ⊢ ((F ‘A) ∈ S → A ∈ S)

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092  ∅c0 1707  dom cdm 2410   ‘cfv 2422

This theorem is referenced by:  reclem1pr 3950  reclem2pr 3951

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

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