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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 6700 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
3 | 2 | eleq1d 2897 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
4 | 1, 3 | mtbiri 329 | . . 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 | eleqtrdi 2923 | 1 ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ∅c0 4291 dom cdm 5555 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 ax-pow 5266 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-dm 5565 df-iota 6314 df-fv 6363 |
This theorem is referenced by: lterpq 10392 ltrnq 10401 reclem2pr 10470 msrrcl 32790 |
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