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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 4248 | . 2 ⊢ (𝐼 ∈ (𝐹‘𝑋) → ¬ (𝐹‘𝑋) = ∅) | |
2 | mptrcl.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | dmmptss 6104 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐴 |
4 | 3 | sseli 3896 | . . 3 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝐴) |
5 | ndmfv 6747 | . . 3 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅) | |
6 | 4, 5 | nsyl4 161 | . 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 1543 ∈ wcel 2110 ∅c0 4237 ↦ cmpt 5135 dom cdm 5551 ‘cfv 6380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-xp 5557 df-rel 5558 df-cnv 5559 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fv 6388 |
This theorem is referenced by: bitsval 15983 subcrcl 17321 initorcl 17496 termorcl 17497 zeroorcl 17498 submrcl 18229 issubg 18543 isnsg 18571 issubrg 19800 issdrg 19839 abvrcl 19857 isobs 20682 islocfin 22414 kgeni 22434 elmptrab 22724 isphtpc 23891 cfili 24165 cfilfcls 24171 plybss 25088 eleenn 26987 neircl 45871 |
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