Theorem ndmfvrcl 3746

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 3745	. . . . 5 |- (-. A e. dom F -> (F` A) = (/))
3	2	eleq1d 1540	. . . 4 |- (-. A e. dom F -> ((F` A) e. S <-> (/) e. S))
4	1, 3	mtbiri 717	. . 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 1558	1 |- ((F` A) e. S -> A e. S)

Syntax hints:  -. wn 2   -> wi 3   = wceq 956   e. wcel 958  (/)c0 2280  dom cdm 3170  ` cfv 3182

This theorem is referenced by:  reclem1pr 5156  reclem2pr 5157

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198

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