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Mirrors > Home > ILE Home > Th. List > preimaf1ofi | GIF version |
Description: The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.) |
Ref | Expression |
---|---|
preimaf1ofi.ss | ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
preimaf1ofi.f | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
preimaf1ofi.c | ⊢ (𝜑 → 𝐶 ∈ Fin) |
Ref | Expression |
---|---|
preimaf1ofi | ⊢ (𝜑 → (◡𝐹 “ 𝐶) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preimaf1ofi.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Fin) | |
2 | preimaf1ofi.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | |
3 | f1ocnv 5380 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
4 | f1of1 5366 | . . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵–1-1→𝐴) | |
5 | 2, 3, 4 | 3syl 17 | . . 3 ⊢ (𝜑 → ◡𝐹:𝐵–1-1→𝐴) |
6 | preimaf1ofi.ss | . . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
7 | f1ores 5382 | . . 3 ⊢ ((◡𝐹:𝐵–1-1→𝐴 ∧ 𝐶 ⊆ 𝐵) → (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶)) | |
8 | 5, 6, 7 | syl2anc 408 | . 2 ⊢ (𝜑 → (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶)) |
9 | f1ofi 6831 | . 2 ⊢ ((𝐶 ∈ Fin ∧ (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶)) → (◡𝐹 “ 𝐶) ∈ Fin) | |
10 | 1, 8, 9 | syl2anc 408 | 1 ⊢ (𝜑 → (◡𝐹 “ 𝐶) ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ⊆ wss 3071 ◡ccnv 4538 ↾ cres 4541 “ cima 4542 –1-1→wf1 5120 –1-1-onto→wf1o 5122 Fincfn 6634 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: fisumss 11164 |
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