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Mirrors > Home > ILE Home > Th. List > dmeq | GIF version |
Description: Equality theorem for domain. (Contributed by NM, 11-Aug-1994.) |
Ref | Expression |
---|---|
dmeq | ⊢ (A = B → dom A = dom B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmss 4477 | . . 3 ⊢ (A ⊆ B → dom A ⊆ dom B) | |
2 | dmss 4477 | . . 3 ⊢ (B ⊆ A → dom B ⊆ dom A) | |
3 | 1, 2 | anim12i 321 | . 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → (dom A ⊆ dom B ∧ dom B ⊆ dom A)) |
4 | eqss 2954 | . 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
5 | eqss 2954 | . 2 ⊢ (dom A = dom B ↔ (dom A ⊆ dom B ∧ dom B ⊆ dom A)) | |
6 | 3, 4, 5 | 3imtr4i 190 | 1 ⊢ (A = B → dom A = dom B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ⊆ wss 2911 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-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-dm 4298 |
This theorem is referenced by: dmeqi 4479 dmeqd 4480 xpid11m 4500 fneq1 4930 eqfnfv2 5209 offval 5661 ofrfval 5662 offval3 5703 smoeq 5846 tfrlemi14d 5888 rdgivallem 5908 rdg0 5914 frec0g 5922 frecsuclem3 5929 frecsuc 5930 ereq1 6049 fundmeng 6223 |
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