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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 4346 | . 2 ⊢ (𝐼 ∈ (𝐹‘𝑋) → ¬ (𝐹‘𝑋) = ∅) | |
2 | mptrcl.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | dmmptss 6263 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐴 |
4 | 3 | sseli 3991 | . . 3 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝐴) |
5 | ndmfv 6942 | . . 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 1537 ∈ wcel 2106 ∅c0 4339 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fv 6571 |
This theorem is referenced by: bitsval 16458 subcrcl 17864 initorcl 18044 termorcl 18045 zeroorcl 18046 submrcl 18828 issubg 19157 isnsg 19186 issubrng 20564 issubrg 20588 issdrg 20806 abvrcl 20831 isobs 21758 mhprcl 22165 islocfin 23541 kgeni 23561 elmptrab 23851 isphtpc 25040 cfili 25316 cfilfcls 25322 plybss 26248 eleenn 28926 neircl 48701 |
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