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Mirrors > Home > ILE Home > Th. List > mptrcl | 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.) (Revised by Jim Kingdon, 27-Mar-2023.) |
Ref | Expression |
---|---|
fvmpt2.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
mptrcl | ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmpt2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | dmmptss 5124 | . 2 ⊢ dom 𝐹 ⊆ 𝐴 |
3 | 1 | funmpt2 5254 | . . . 4 ⊢ Fun 𝐹 |
4 | funrel 5232 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ Rel 𝐹 |
6 | relelfvdm 5546 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐼 ∈ (𝐹‘𝑋)) → 𝑋 ∈ dom 𝐹) | |
7 | 5, 6 | mpan 424 | . 2 ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ dom 𝐹) |
8 | 2, 7 | sselid 3153 | 1 ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ↦ cmpt 4063 dom cdm 4625 Rel wrel 4630 Fun wfun 5209 ‘cfv 5215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fv 5223 |
This theorem is referenced by: submrcl 12794 issubg 12964 isnsg 12993 issubrg 13280 psmetdmdm 13695 psmetf 13696 psmet0 13698 psmettri2 13699 psmetres2 13704 |
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