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Mirrors > Home > ILE Home > Th. List > dmmptg | GIF version |
Description: The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.) |
Ref | Expression |
---|---|
dmmptg | ⊢ (∀x ∈ A B ∈ 𝑉 → dom (x ∈ A ↦ B) = A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2560 | . . . 4 ⊢ (B ∈ 𝑉 → B ∈ V) | |
2 | 1 | ralimi 2378 | . . 3 ⊢ (∀x ∈ A B ∈ 𝑉 → ∀x ∈ A B ∈ V) |
3 | rabid2 2480 | . . 3 ⊢ (A = {x ∈ A ∣ B ∈ V} ↔ ∀x ∈ A B ∈ V) | |
4 | 2, 3 | sylibr 137 | . 2 ⊢ (∀x ∈ A B ∈ 𝑉 → A = {x ∈ A ∣ B ∈ V}) |
5 | eqid 2037 | . . 3 ⊢ (x ∈ A ↦ B) = (x ∈ A ↦ B) | |
6 | 5 | dmmpt 4759 | . 2 ⊢ dom (x ∈ A ↦ B) = {x ∈ A ∣ B ∈ V} |
7 | 4, 6 | syl6reqr 2088 | 1 ⊢ (∀x ∈ A B ∈ 𝑉 → dom (x ∈ A ↦ B) = A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 ∀wral 2300 {crab 2304 Vcvv 2551 ↦ 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: resfunexg 5325 rdgtfr 5901 rdgruledefgg 5902 |
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