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Mirrors > Home > ILE Home > Th. List > dminss | GIF version |
Description: An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.) |
Ref | Expression |
---|---|
dminss | ⊢ (dom 𝑅 ∩ A) ⊆ (◡𝑅 “ (𝑅 “ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 1479 | . . . . . . 7 ⊢ ((x ∈ A ∧ x𝑅y) → ∃x(x ∈ A ∧ x𝑅y)) | |
2 | 1 | ancoms 255 | . . . . . 6 ⊢ ((x𝑅y ∧ x ∈ A) → ∃x(x ∈ A ∧ x𝑅y)) |
3 | vex 2554 | . . . . . . 7 ⊢ y ∈ V | |
4 | 3 | elima2 4617 | . . . . . 6 ⊢ (y ∈ (𝑅 “ A) ↔ ∃x(x ∈ A ∧ x𝑅y)) |
5 | 2, 4 | sylibr 137 | . . . . 5 ⊢ ((x𝑅y ∧ x ∈ A) → y ∈ (𝑅 “ A)) |
6 | simpl 102 | . . . . . 6 ⊢ ((x𝑅y ∧ x ∈ A) → x𝑅y) | |
7 | vex 2554 | . . . . . . 7 ⊢ x ∈ V | |
8 | 3, 7 | brcnv 4461 | . . . . . 6 ⊢ (y◡𝑅x ↔ x𝑅y) |
9 | 6, 8 | sylibr 137 | . . . . 5 ⊢ ((x𝑅y ∧ x ∈ A) → y◡𝑅x) |
10 | 5, 9 | jca 290 | . . . 4 ⊢ ((x𝑅y ∧ x ∈ A) → (y ∈ (𝑅 “ A) ∧ y◡𝑅x)) |
11 | 10 | eximi 1488 | . . 3 ⊢ (∃y(x𝑅y ∧ x ∈ A) → ∃y(y ∈ (𝑅 “ A) ∧ y◡𝑅x)) |
12 | 7 | eldm 4475 | . . . . 5 ⊢ (x ∈ dom 𝑅 ↔ ∃y x𝑅y) |
13 | 12 | anbi1i 431 | . . . 4 ⊢ ((x ∈ dom 𝑅 ∧ x ∈ A) ↔ (∃y x𝑅y ∧ x ∈ A)) |
14 | elin 3120 | . . . 4 ⊢ (x ∈ (dom 𝑅 ∩ A) ↔ (x ∈ dom 𝑅 ∧ x ∈ A)) | |
15 | 19.41v 1779 | . . . 4 ⊢ (∃y(x𝑅y ∧ x ∈ A) ↔ (∃y x𝑅y ∧ x ∈ A)) | |
16 | 13, 14, 15 | 3bitr4i 201 | . . 3 ⊢ (x ∈ (dom 𝑅 ∩ A) ↔ ∃y(x𝑅y ∧ x ∈ A)) |
17 | 7 | elima2 4617 | . . 3 ⊢ (x ∈ (◡𝑅 “ (𝑅 “ A)) ↔ ∃y(y ∈ (𝑅 “ A) ∧ y◡𝑅x)) |
18 | 11, 16, 17 | 3imtr4i 190 | . 2 ⊢ (x ∈ (dom 𝑅 ∩ A) → x ∈ (◡𝑅 “ (𝑅 “ A))) |
19 | 18 | ssriv 2943 | 1 ⊢ (dom 𝑅 ∩ A) ⊆ (◡𝑅 “ (𝑅 “ A)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∃wex 1378 ∈ wcel 1390 ∩ cin 2910 ⊆ wss 2911 class class class wbr 3755 ◡ccnv 4287 dom cdm 4288 “ cima 4291 |
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-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-xp 4294 df-cnv 4296 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 |
This theorem is referenced by: (None) |
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