|   | Mathbox for Scott Fenton | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > elintfv | Structured version Visualization version GIF version | ||
| Description: Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.) | 
| Ref | Expression | 
|---|---|
| elintfv.1 | ⊢ 𝑋 ∈ V | 
| Ref | Expression | 
|---|---|
| elintfv | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elintfv.1 | . . 3 ⊢ 𝑋 ∈ V | |
| 2 | 1 | elint 4952 | . 2 ⊢ (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧)) | 
| 3 | fvelimab 6981 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧)) | |
| 4 | 3 | imbi1d 341 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))) | 
| 5 | r19.23v 3183 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)) | |
| 6 | 4, 5 | bitr4di 289 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))) | 
| 7 | 6 | albidv 1920 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))) | 
| 8 | ralcom4 3286 | . . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)) | |
| 9 | eqcom 2744 | . . . . . . . 8 ⊢ ((𝐹‘𝑦) = 𝑧 ↔ 𝑧 = (𝐹‘𝑦)) | |
| 10 | 9 | imbi1i 349 | . . . . . . 7 ⊢ (((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ (𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧)) | 
| 11 | 10 | albii 1819 | . . . . . 6 ⊢ (∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑧(𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧)) | 
| 12 | fvex 6919 | . . . . . . 7 ⊢ (𝐹‘𝑦) ∈ V | |
| 13 | eleq2 2830 | . . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑋 ∈ 𝑧 ↔ 𝑋 ∈ (𝐹‘𝑦))) | |
| 14 | 12, 13 | ceqsalv 3521 | . . . . . 6 ⊢ (∀𝑧(𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧) ↔ 𝑋 ∈ (𝐹‘𝑦)) | 
| 15 | 11, 14 | bitri 275 | . . . . 5 ⊢ (∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ 𝑋 ∈ (𝐹‘𝑦)) | 
| 16 | 15 | ralbii 3093 | . . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)) | 
| 17 | 8, 16 | bitr3i 277 | . . 3 ⊢ (∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)) | 
| 18 | 7, 17 | bitrdi 287 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))) | 
| 19 | 2, 18 | bitrid 283 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 ∩ cint 4946 “ cima 5688 Fn wfn 6556 ‘cfv 6561 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-id 5578 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-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |