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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 4340 | . 2 ⊢ (𝐼 ∈ (𝐹‘𝑋) → ¬ (𝐹‘𝑋) = ∅) | |
| 2 | mptrcl.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 3 | 2 | dmmptss 6261 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐴 |
| 4 | 3 | sseli 3979 | . . 3 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝐴) |
| 5 | ndmfv 6941 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅) | |
| 6 | 4, 5 | nsyl4 158 | . 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 1540 ∈ wcel 2108 ∅c0 4333 ↦ cmpt 5225 dom cdm 5685 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: bitsval 16461 subcrcl 17860 initorcl 18035 termorcl 18036 zeroorcl 18037 submrcl 18815 issubg 19144 isnsg 19173 issubrng 20547 issubrg 20571 issdrg 20789 abvrcl 20814 isobs 21740 mhprcl 22147 islocfin 23525 kgeni 23545 elmptrab 23835 isphtpc 25026 cfili 25302 cfilfcls 25308 plybss 26233 eleenn 28911 neircl 48802 |
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