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Mirrors > Home > ILE Home > Th. List > rnin | GIF version |
Description: The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) |
Ref | Expression |
---|---|
rnin | ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvin 4751 | . . . 4 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵) | |
2 | 1 | dmeqi 4554 | . . 3 ⊢ dom ◡(𝐴 ∩ 𝐵) = dom (◡𝐴 ∩ ◡𝐵) |
3 | dmin 4561 | . . 3 ⊢ dom (◡𝐴 ∩ ◡𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵) | |
4 | 2, 3 | eqsstri 3029 | . 2 ⊢ dom ◡(𝐴 ∩ 𝐵) ⊆ (dom ◡𝐴 ∩ dom ◡𝐵) |
5 | df-rn 4374 | . 2 ⊢ ran (𝐴 ∩ 𝐵) = dom ◡(𝐴 ∩ 𝐵) | |
6 | df-rn 4374 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | df-rn 4374 | . . 3 ⊢ ran 𝐵 = dom ◡𝐵 | |
8 | 6, 7 | ineq12i 3165 | . 2 ⊢ (ran 𝐴 ∩ ran 𝐵) = (dom ◡𝐴 ∩ dom ◡𝐵) |
9 | 4, 5, 8 | 3sstr4i 3038 | 1 ⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∩ cin 2972 ⊆ wss 2973 ◡ccnv 4362 dom cdm 4363 ran crn 4364 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: inimass 4760 |
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