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Mirrors > Home > MPE Home > Th. List > mptrcl | Structured version Visualization version GIF version |
Description: Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) |
Ref | Expression |
---|---|
mptrcl.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
mptrcl | ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 3879 | . 2 ⊢ (𝐼 ∈ (𝐹‘𝑋) → ¬ (𝐹‘𝑋) = ∅) | |
2 | mptrcl.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | dmmptss 5548 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐴 |
4 | 3 | sseli 3564 | . . 3 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝐴) |
5 | ndmfv 6128 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅) | |
6 | 4, 5 | nsyl4 155 | . 2 ⊢ (¬ (𝐹‘𝑋) = ∅ → 𝑋 ∈ 𝐴) |
7 | 1, 6 | syl 17 | 1 ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ∅c0 3874 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fv 5812 |
This theorem is referenced by: initorcl 16467 termorcl 16468 zeroorcl 16469 issubrg 18603 elmptrab 21440 |
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