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Mirrors > Home > ILE Home > Th. List > dmmptss | GIF version |
Description: The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.) |
Ref | Expression |
---|---|
dmmpt2.1 | ⊢ 𝐹 = (x ∈ A ↦ B) |
Ref | Expression |
---|---|
dmmptss | ⊢ dom 𝐹 ⊆ A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmpt2.1 | . . 3 ⊢ 𝐹 = (x ∈ A ↦ B) | |
2 | 1 | dmmpt 4759 | . 2 ⊢ dom 𝐹 = {x ∈ A ∣ B ∈ V} |
3 | ssrab2 3019 | . 2 ⊢ {x ∈ A ∣ B ∈ V} ⊆ A | |
4 | 2, 3 | eqsstri 2969 | 1 ⊢ dom 𝐹 ⊆ A |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∈ wcel 1390 {crab 2304 Vcvv 2551 ⊆ wss 2911 ↦ cmpt 3809 dom cdm 4288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-mpt 3811 df-xp 4294 df-rel 4295 df-cnv 4296 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 |
This theorem is referenced by: fvmptssdm 5198 mptexg 5329 dmmpt2ssx 5767 tposssxp 5805 |
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