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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 6218 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
3 | 2 | eleq1d 2686 | . . . 4 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ((𝐹‘𝐴) ∈ 𝑆 ↔ ∅ ∈ 𝑆)) |
4 | 1, 3 | mtbiri 317 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ (𝐹‘𝐴) ∈ 𝑆) |
5 | 4 | con4i 113 | . 2 ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ dom 𝐹) |
6 | ndmfvrcl.1 | . 2 ⊢ dom 𝐹 = 𝑆 | |
7 | 5, 6 | syl6eleq 2711 | 1 ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ∅c0 3915 dom cdm 5114 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 ax-pow 4843 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 |
This theorem is referenced by: lterpq 9792 ltrnq 9801 reclem2pr 9870 msrrcl 31440 |
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