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Mirrors > Home > ILE Home > Th. List > relcnvfi | GIF version |
Description: If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.) |
Ref | Expression |
---|---|
relcnvfi | ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel2 4915 | . . . . 5 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | 1 | biimpi 119 | . . . 4 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴) |
3 | 2 | adantr 271 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡◡𝐴 = 𝐴) |
4 | simpr 109 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
5 | 3, 4 | eqeltrd 2171 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡◡𝐴 ∈ Fin) |
6 | relcnv 4843 | . . . 4 ⊢ Rel ◡𝐴 | |
7 | cnvexg 5002 | . . . 4 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ V) | |
8 | cnven 6605 | . . . 4 ⊢ ((Rel ◡𝐴 ∧ ◡𝐴 ∈ V) → ◡𝐴 ≈ ◡◡𝐴) | |
9 | 6, 7, 8 | sylancr 406 | . . 3 ⊢ (𝐴 ∈ Fin → ◡𝐴 ≈ ◡◡𝐴) |
10 | 9 | adantl 272 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ≈ ◡◡𝐴) |
11 | enfii 6670 | . 2 ⊢ ((◡◡𝐴 ∈ Fin ∧ ◡𝐴 ≈ ◡◡𝐴) → ◡𝐴 ∈ Fin) | |
12 | 5, 10, 11 | syl2anc 404 | 1 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 Vcvv 2633 class class class wbr 3867 ◡ccnv 4466 Rel wrel 4472 ≈ cen 6535 Fincfn 6537 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-1st 5949 df-2nd 5950 df-er 6332 df-en 6538 df-fin 6540 |
This theorem is referenced by: funrnfi 6731 fsumcnv 10980 |
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