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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 5100 | . . . . 5 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | 1 | biimpi 120 | . . . 4 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴) |
3 | 2 | adantr 276 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡◡𝐴 = 𝐴) |
4 | simpr 110 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
5 | 3, 4 | eqeltrd 2266 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡◡𝐴 ∈ Fin) |
6 | relcnv 5027 | . . . 4 ⊢ Rel ◡𝐴 | |
7 | cnvexg 5187 | . . . 4 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ V) | |
8 | cnven 6838 | . . . 4 ⊢ ((Rel ◡𝐴 ∧ ◡𝐴 ∈ V) → ◡𝐴 ≈ ◡◡𝐴) | |
9 | 6, 7, 8 | sylancr 414 | . . 3 ⊢ (𝐴 ∈ Fin → ◡𝐴 ≈ ◡◡𝐴) |
10 | 9 | adantl 277 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ≈ ◡◡𝐴) |
11 | enfii 6906 | . 2 ⊢ ((◡◡𝐴 ∈ Fin ∧ ◡𝐴 ≈ ◡◡𝐴) → ◡𝐴 ∈ Fin) | |
12 | 5, 10, 11 | syl2anc 411 | 1 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 Vcvv 2752 class class class wbr 4021 ◡ccnv 4646 Rel wrel 4652 ≈ cen 6768 Fincfn 6770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-1st 6169 df-2nd 6170 df-er 6563 df-en 6771 df-fin 6773 |
This theorem is referenced by: funrnfi 6975 fsumcnv 11486 fprodcnv 11674 |
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