| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > fiss | Structured version Visualization version 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 | sstr2 3990 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝑦 → 𝐴 ⊆ 𝑦)) | |
| 2 | 1 | adantl 481 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝐵 ⊆ 𝑦 → 𝐴 ⊆ 𝑦)) |
| 3 | 2 | anim1d 611 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → ((𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦) → (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦))) |
| 4 | 3 | ss2abdv 4066 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)} ⊆ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)}) |
| 5 | intss 4969 | . . 3 ⊢ ({𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)} ⊆ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)} → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)} ⊆ ∩ {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)}) | |
| 6 | 4, 5 | syl 17 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)} ⊆ ∩ {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)}) |
| 7 | ssexg 5323 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) | |
| 8 | 7 | ancoms 458 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V) |
| 9 | dffi2 9463 | . . 3 ⊢ (𝐴 ∈ V → (fi‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)}) | |
| 10 | 8, 9 | syl 17 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)}) |
| 11 | dffi2 9463 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (fi‘𝐵) = ∩ {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)}) | |
| 12 | 11 | adantr 480 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐵) = ∩ {𝑦 ∣ (𝐵 ⊆ 𝑦 ∧ ∀𝑥 ∈ 𝑦 ∀𝑧 ∈ 𝑦 (𝑥 ∩ 𝑧) ∈ 𝑦)}) |
| 13 | 6, 10, 12 | 3sstr4d 4039 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∩ cint 4946 ‘cfv 6561 ficfi 9450 |
| 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-pow 5365 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-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 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-2o 8507 df-en 8986 df-fin 8989 df-fi 9451 |
| This theorem is referenced by: fipwuni 9466 elfiun 9470 tgfiss 22998 ordtbas 23200 leordtval2 23220 lecldbas 23227 2ndcsb 23457 ptbasfi 23589 fclscmpi 24037 prdsxmslem2 24542 |
| Copyright terms: Public domain | W3C validator |