Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ssdomg | GIF version |
Description: A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
ssdomg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 4037 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) | |
2 | simpr 109 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
3 | f1oi 5373 | . . . . . . . . . 10 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
4 | dff1o3 5341 | . . . . . . . . . 10 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 ↔ (( I ↾ 𝐴):𝐴–onto→𝐴 ∧ Fun ◡( I ↾ 𝐴))) | |
5 | 3, 4 | mpbi 144 | . . . . . . . . 9 ⊢ (( I ↾ 𝐴):𝐴–onto→𝐴 ∧ Fun ◡( I ↾ 𝐴)) |
6 | 5 | simpli 110 | . . . . . . . 8 ⊢ ( I ↾ 𝐴):𝐴–onto→𝐴 |
7 | fof 5315 | . . . . . . . 8 ⊢ (( I ↾ 𝐴):𝐴–onto→𝐴 → ( I ↾ 𝐴):𝐴⟶𝐴) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ ( I ↾ 𝐴):𝐴⟶𝐴 |
9 | fss 5254 | . . . . . . 7 ⊢ ((( I ↾ 𝐴):𝐴⟶𝐴 ∧ 𝐴 ⊆ 𝐵) → ( I ↾ 𝐴):𝐴⟶𝐵) | |
10 | 8, 9 | mpan 420 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴):𝐴⟶𝐵) |
11 | funi 5125 | . . . . . . . 8 ⊢ Fun I | |
12 | cnvi 4913 | . . . . . . . . 9 ⊢ ◡ I = I | |
13 | 12 | funeqi 5114 | . . . . . . . 8 ⊢ (Fun ◡ I ↔ Fun I ) |
14 | 11, 13 | mpbir 145 | . . . . . . 7 ⊢ Fun ◡ I |
15 | funres11 5165 | . . . . . . 7 ⊢ (Fun ◡ I → Fun ◡( I ↾ 𝐴)) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ Fun ◡( I ↾ 𝐴) |
17 | 10, 16 | jctir 311 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (( I ↾ 𝐴):𝐴⟶𝐵 ∧ Fun ◡( I ↾ 𝐴))) |
18 | df-f1 5098 | . . . . 5 ⊢ (( I ↾ 𝐴):𝐴–1-1→𝐵 ↔ (( I ↾ 𝐴):𝐴⟶𝐵 ∧ Fun ◡( I ↾ 𝐴))) | |
19 | 17, 18 | sylibr 133 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴):𝐴–1-1→𝐵) |
20 | 19 | adantr 274 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → ( I ↾ 𝐴):𝐴–1-1→𝐵) |
21 | f1dom2g 6618 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉 ∧ ( I ↾ 𝐴):𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) | |
22 | 1, 2, 20, 21 | syl3anc 1201 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ≼ 𝐵) |
23 | 22 | expcom 115 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 Vcvv 2660 ⊆ wss 3041 class class class wbr 3899 I cid 4180 ◡ccnv 4508 ↾ cres 4511 Fun wfun 5087 ⟶wf 5089 –1-1→wf1 5090 –onto→wfo 5091 –1-1-onto→wf1o 5092 ≼ cdom 6601 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-dom 6604 |
This theorem is referenced by: cnvct 6671 ssct 6680 xpdom3m 6696 0domg 6699 mapdom1g 6709 phplem4dom 6724 nndomo 6726 phpm 6727 fict 6730 domfiexmid 6740 infnfi 6757 exmidfodomrlemr 7026 exmidfodomrlemrALT 7027 fihashss 10530 phicl2 11817 phibnd 11820 qnnen 11871 pw1dom2 13117 sbthom 13148 |
Copyright terms: Public domain | W3C validator |