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Mirrors > Home > ILE Home > Th. List > cnvct | GIF version |
Description: If a set is dominated by ω, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
Ref | Expression |
---|---|
cnvct | ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4925 | . . . 4 ⊢ Rel ◡𝐴 | |
2 | ctex 6655 | . . . . 5 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
3 | cnvexg 5084 | . . . . 5 ⊢ (𝐴 ∈ V → ◡𝐴 ∈ V) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐴 ≼ ω → ◡𝐴 ∈ V) |
5 | cnven 6710 | . . . 4 ⊢ ((Rel ◡𝐴 ∧ ◡𝐴 ∈ V) → ◡𝐴 ≈ ◡◡𝐴) | |
6 | 1, 4, 5 | sylancr 411 | . . 3 ⊢ (𝐴 ≼ ω → ◡𝐴 ≈ ◡◡𝐴) |
7 | cnvcnvss 5001 | . . . 4 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
8 | ssdomg 6680 | . . . 4 ⊢ (𝐴 ∈ V → (◡◡𝐴 ⊆ 𝐴 → ◡◡𝐴 ≼ 𝐴)) | |
9 | 2, 7, 8 | mpisyl 1423 | . . 3 ⊢ (𝐴 ≼ ω → ◡◡𝐴 ≼ 𝐴) |
10 | endomtr 6692 | . . 3 ⊢ ((◡𝐴 ≈ ◡◡𝐴 ∧ ◡◡𝐴 ≼ 𝐴) → ◡𝐴 ≼ 𝐴) | |
11 | 6, 9, 10 | syl2anc 409 | . 2 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ 𝐴) |
12 | domtr 6687 | . 2 ⊢ ((◡𝐴 ≼ 𝐴 ∧ 𝐴 ≼ ω) → ◡𝐴 ≼ ω) | |
13 | 11, 12 | mpancom 419 | 1 ⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 class class class wbr 3937 ωcom 4512 ◡ccnv 4546 Rel wrel 4552 ≈ cen 6640 ≼ cdom 6641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1st 6046 df-2nd 6047 df-en 6643 df-dom 6644 |
This theorem is referenced by: (None) |
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