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| Mirrors > Home > MPE Home > Th. List > domfi | Structured version Visualization version GIF version | ||
| Description: A set dominated by a finite set is finite. (Contributed by NM, 23-Mar-2006.) (Revised by Mario Carneiro, 12-Mar-2015.) |
| Ref | Expression |
|---|---|
| domfi | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | domeng 9003 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐵 ≼ 𝐴 ↔ ∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴))) | |
| 2 | ssfi 9213 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin) | |
| 3 | 2 | adantrl 716 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝑥 ∈ Fin) |
| 4 | enfii 9226 | . . . . . . 7 ⊢ ((𝑥 ∈ Fin ∧ 𝐵 ≈ 𝑥) → 𝐵 ∈ Fin) | |
| 5 | 4 | adantrr 717 | . . . . . 6 ⊢ ((𝑥 ∈ Fin ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝐵 ∈ Fin) |
| 6 | 3, 5 | sylancom 588 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) → 𝐵 ∈ Fin) |
| 7 | 6 | ex 412 | . . . 4 ⊢ (𝐴 ∈ Fin → ((𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → 𝐵 ∈ Fin)) |
| 8 | 7 | exlimdv 1933 | . . 3 ⊢ (𝐴 ∈ Fin → (∃𝑥(𝐵 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐴) → 𝐵 ∈ Fin)) |
| 9 | 1, 8 | sylbid 240 | . 2 ⊢ (𝐴 ∈ Fin → (𝐵 ≼ 𝐴 → 𝐵 ∈ Fin)) |
| 10 | 9 | imp 406 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 ≈ cen 8982 ≼ cdom 8983 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-en 8986 df-dom 8987 df-fin 8989 |
| This theorem is referenced by: domtrfi 9233 domtrfir 9234 sdomdomtrfi 9241 php3 9249 onomeneq 9265 xpfir 9300 findcard3 9318 fofi 9351 fodomfir 9368 dmfi 9375 sdom2en01 10342 isfin1-2 10425 fin67 10435 fin1a2lem9 10448 gchdju1 10696 hashdomi 14419 symggen 19488 cmpsub 23408 ufinffr 23937 alexsubALT 24059 ovolicc2lem4 25555 aannenlem1 26370 madefi 27950 ffsrn 32740 locfinreflem 33839 lindsenlbs 37622 harinf 43046 kelac2lem 43076 disjinfi 45197 |
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