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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 5476 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
4 | f1of1 5462 | . . . 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 5478 | . . 3 ⊢ ((◡𝐹:𝐵–1-1→𝐴 ∧ 𝐶 ⊆ 𝐵) → (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶)) | |
8 | 5, 6, 7 | syl2anc 411 | . 2 ⊢ (𝜑 → (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶)) |
9 | f1ofi 6944 | . 2 ⊢ ((𝐶 ∈ Fin ∧ (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶)) → (◡𝐹 “ 𝐶) ∈ Fin) | |
10 | 1, 8, 9 | syl2anc 411 | 1 ⊢ (𝜑 → (◡𝐹 “ 𝐶) ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ⊆ wss 3131 ◡ccnv 4627 ↾ cres 4630 “ cima 4631 –1-1→wf1 5215 –1-1-onto→wf1o 5217 Fincfn 6742 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-er 6537 df-en 6743 df-fin 6745 |
This theorem is referenced by: fisumss 11402 fprodssdc 11600 |
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