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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 4993 | . 2 ⊢ dom 𝐹 ⊆ 𝐴 |
3 | 1 | funmpt2 5120 | . . . 4 ⊢ Fun 𝐹 |
4 | funrel 5098 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | ax-mp 7 | . . 3 ⊢ Rel 𝐹 |
6 | relelfvdm 5407 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐼 ∈ (𝐹‘𝑋)) → 𝑋 ∈ dom 𝐹) | |
7 | 5, 6 | mpan 418 | . 2 ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ dom 𝐹) |
8 | 2, 7 | sseldi 3061 | 1 ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 ∈ wcel 1463 ↦ cmpt 3949 dom cdm 4499 Rel wrel 4504 Fun wfun 5075 ‘cfv 5081 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fv 5089 |
This theorem is referenced by: psmetdmdm 12313 psmetf 12314 psmet0 12316 psmettri2 12317 psmetres2 12322 |
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