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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 110 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
2 | sspwb 4213 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) | |
3 | ssrin 3360 | . . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐵 ∩ Fin)) | |
4 | 2, 3 | sylbi 121 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝒫 𝐴 ∩ Fin) ⊆ (𝒫 𝐵 ∩ Fin)) |
5 | ssrexv 3220 | . . . 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 2740 | . . . 4 ⊢ 𝑟 ∈ V | |
8 | simpl 109 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ 𝑉) | |
9 | 8, 1 | ssexd 4140 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V) |
10 | elfi 6964 | . . . 4 ⊢ ((𝑟 ∈ V ∧ 𝐴 ∈ V) → (𝑟 ∈ (fi‘𝐴) ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑟 = ∩ 𝑥)) | |
11 | 7, 9, 10 | sylancr 414 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝑟 ∈ (fi‘𝐴) ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑟 = ∩ 𝑥)) |
12 | elfi 6964 | . . . . 5 ⊢ ((𝑟 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝑟 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥)) | |
13 | 7, 12 | mpan 424 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑟 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥)) |
14 | 13 | adantr 276 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝑟 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝑟 = ∩ 𝑥)) |
15 | 6, 11, 14 | 3imtr4d 203 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝑟 ∈ (fi‘𝐴) → 𝑟 ∈ (fi‘𝐵))) |
16 | 15 | ssrdv 3161 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 Vcvv 2737 ∩ cin 3128 ⊆ wss 3129 𝒫 cpw 3574 ∩ cint 3842 ‘cfv 5212 Fincfn 6734 ficfi 6961 |
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-in1 614 ax-in2 615 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-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-iinf 4584 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-er 6529 df-en 6735 df-fin 6737 df-fi 6962 |
This theorem is referenced by: (None) |
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