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Mirrors > Home > ILE Home > Th. List > f1finf1o | GIF version |
Description: Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) |
Ref | Expression |
---|---|
f1finf1o | ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1→𝐵) | |
2 | simplr 525 | . . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐵 ∈ Fin) | |
3 | f1rn 5402 | . . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵) | |
4 | 3 | adantl 275 | . . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ⊆ 𝐵) |
5 | f1fn 5403 | . . . . . . . . 9 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | |
6 | fnima 5314 | . . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) | |
7 | 5, 6 | syl 14 | . . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 “ 𝐴) = ran 𝐹) |
8 | 7 | adantl 275 | . . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 “ 𝐴) = ran 𝐹) |
9 | ssid 3167 | . . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴 | |
10 | 9 | a1i 9 | . . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ⊆ 𝐴) |
11 | simpll 524 | . . . . . . . . 9 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≈ 𝐵) | |
12 | enfii 6848 | . . . . . . . . 9 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) | |
13 | 2, 11, 12 | syl2anc 409 | . . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ∈ Fin) |
14 | f1imaeng 6766 | . . . . . . . 8 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ⊆ 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 “ 𝐴) ≈ 𝐴) | |
15 | 1, 10, 13, 14 | syl3anc 1233 | . . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 “ 𝐴) ≈ 𝐴) |
16 | 8, 15 | eqbrtrrd 4011 | . . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ≈ 𝐴) |
17 | entr 6758 | . . . . . 6 ⊢ ((ran 𝐹 ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → ran 𝐹 ≈ 𝐵) | |
18 | 16, 11, 17 | syl2anc 409 | . . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ≈ 𝐵) |
19 | fisseneq 6905 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊆ 𝐵 ∧ ran 𝐹 ≈ 𝐵) → ran 𝐹 = 𝐵) | |
20 | 2, 4, 18, 19 | syl3anc 1233 | . . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 = 𝐵) |
21 | dff1o5 5449 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵)) | |
22 | 1, 20, 21 | sylanbrc 415 | . . 3 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1-onto→𝐵) |
23 | 22 | ex 114 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→𝐵)) |
24 | f1of1 5439 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵) | |
25 | 23, 24 | impbid1 141 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 ⊆ wss 3121 class class class wbr 3987 ran crn 4610 “ cima 4612 Fn wfn 5191 –1-1→wf1 5193 –1-1-onto→wf1o 5195 ≈ cen 6712 Fincfn 6714 |
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-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-1o 6392 df-er 6509 df-en 6715 df-fin 6717 |
This theorem is referenced by: iseqf1olemqf1o 10436 crth 12165 eulerthlemh 12172 pwf1oexmid 13992 |
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