Theorem ndmfvrcl 3753

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

Hypotheses

Ref	Expression
ndmfvrcl.1	|- dom F = S
ndmfvrcl.2	|- -. (/) e. S

Assertion

Ref	Expression
ndmfvrcl	|- ((F` A) e. S -> A e. S)

Proof of Theorem ndmfvrcl

Step	Hyp	Ref	Expression
1		ndmfvrcl.2	. . . 4 |- -. (/) e. S
2		ndmfv 3752	. . . . 5 |- (-. A e. dom F -> (F` A) = (/))
3	2	eleq1d 1543	. . . 4 |- (-. A e. dom F -> ((F` A) e. S <-> (/) e. S))
4	1, 3	mtbiri 719	. . 3 |- (-. A e. dom F -> -. (F` A) e. S)
5	4	a3i 74	. 2 |- ((F` A) e. S -> A e. dom F)
6		ndmfvrcl.1	. 2 |- dom F = S
7	5, 6	syl6eleq 1561	1 |- ((F` A) e. S -> A e. S)

Syntax hints:  -. wn 2   -> wi 3   = wceq 958   e. wcel 960  (/)c0 2284  dom cdm 3177  ` cfv 3189

This theorem is referenced by:  reclem1pr 5175  reclem2pr 5176

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-pow 2749  ax-pr 2786

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-xp 3191  df-cnv 3193  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fv 3205

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