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Mirrors > Home > ILE Home > Th. List > fiss | GIF version |
Description: Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
fiss | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
2 | sspwb 4146 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) | |
3 | ssrin 3306 | . . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐵 ∩ Fin)) | |
4 | 2, 3 | sylbi 120 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐵 ∩ Fin)) |
5 | ssrexv 3167 | . . . 4 ⊢ ((𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐵 ∩ Fin) → (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑟 = ∩ 𝑥 → ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥)) | |
6 | 1, 4, 5 | 3syl 17 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑟 = ∩ 𝑥 → ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥)) |
7 | vex 2692 | . . . 4 ⊢ 𝑟 ∈ V | |
8 | simpl 108 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ 𝑉) | |
9 | 8, 1 | ssexd 4076 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V) |
10 | elfi 6867 | . . . 4 ⊢ ((𝑟 ∈ V ∧ 𝐴 ∈ V) → (𝑟 ∈ (fi‘𝐴) ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑟 = ∩ 𝑥)) | |
11 | 7, 9, 10 | sylancr 411 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝑟 ∈ (fi‘𝐴) ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑟 = ∩ 𝑥)) |
12 | elfi 6867 | . . . . 5 ⊢ ((𝑟 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝑟 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥)) | |
13 | 7, 12 | mpan 421 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑟 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥)) |
14 | 13 | adantr 274 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝑟 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥)) |
15 | 6, 11, 14 | 3imtr4d 202 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝑟 ∈ (fi‘𝐴) → 𝑟 ∈ (fi‘𝐵))) |
16 | 15 | ssrdv 3108 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∃wrex 2418 Vcvv 2689 ∩ cin 3075 ⊆ wss 3076 𝒫 cpw 3515 ∩ cint 3779 ‘cfv 5131 Fincfn 6642 ficfi 6864 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-er 6437 df-en 6643 df-fin 6645 df-fi 6865 |
This theorem is referenced by: (None) |
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