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Mirrors > Home > MPE Home > Th. List > ndmfvrcl | Structured version Visualization version GIF version |
Description: Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.) |
Ref | Expression |
---|---|
ndmfvrcl.1 | ⊢ dom 𝐹 = 𝑆 |
ndmfvrcl.2 | ⊢ ¬ ∅ ∈ 𝑆 |
Ref | Expression |
---|---|
ndmfvrcl | ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndmfvrcl.2 | . . . 4 ⊢ ¬ ∅ ∈ 𝑆 | |
2 | ndmfv 6529 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
3 | 2 | eleq1d 2851 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
4 | 1, 3 | mtbiri 319 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ (𝐹‘𝐴) ∈ 𝑆) |
5 | 4 | con4i 114 | . 2 ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ dom 𝐹) |
6 | ndmfvrcl.1 | . 2 ⊢ dom 𝐹 = 𝑆 | |
7 | 5, 6 | syl6eleq 2877 | 1 ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1507 ∈ wcel 2050 ∅c0 4179 dom cdm 5407 ‘cfv 6188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 ax-nul 5067 ax-pow 5119 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-dm 5417 df-iota 6152 df-fv 6196 |
This theorem is referenced by: lterpq 10190 ltrnq 10199 reclem2pr 10268 msrrcl 32307 |
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