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| Mirrors > Home > MPE Home > Th. List > resf1st | Structured version Visualization version GIF version | ||
| Description: Value of the functor restriction operator on objects. (Contributed by Mario Carneiro, 6-Jan-2017.) |
| Ref | Expression |
|---|---|
| resf1st.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
| resf1st.h | ⊢ (𝜑 → 𝐻 ∈ 𝑊) |
| resf1st.s | ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) |
| Ref | Expression |
|---|---|
| resf1st | ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | resf1st.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
| 2 | resf1st.h | . . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝑊) | |
| 3 | 1, 2 | resfval 17937 | . . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = 〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) |
| 4 | 3 | fveq2d 6910 | . 2 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = (1st ‘〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉)) |
| 5 | fvex 6919 | . . . 4 ⊢ (1st ‘𝐹) ∈ V | |
| 6 | 5 | resex 6047 | . . 3 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V |
| 7 | dmexg 7923 | . . . 4 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V) | |
| 8 | mptexg 7241 | . . . 4 ⊢ (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V) | |
| 9 | 2, 7, 8 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V) |
| 10 | op1stg 8026 | . . 3 ⊢ ((((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V) → (1st ‘〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) = ((1st ‘𝐹) ↾ dom dom 𝐻)) | |
| 11 | 6, 9, 10 | sylancr 587 | . 2 ⊢ (𝜑 → (1st ‘〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) = ((1st ‘𝐹) ↾ dom dom 𝐻)) |
| 12 | resf1st.s | . . . . . 6 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
| 13 | 12 | fndmd 6673 | . . . . 5 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
| 14 | 13 | dmeqd 5916 | . . . 4 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆)) |
| 15 | dmxpid 5941 | . . . 4 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
| 16 | 14, 15 | eqtrdi 2793 | . . 3 ⊢ (𝜑 → dom dom 𝐻 = 𝑆) |
| 17 | 16 | reseq2d 5997 | . 2 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻) = ((1st ‘𝐹) ↾ 𝑆)) |
| 18 | 4, 11, 17 | 3eqtrd 2781 | 1 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ↦ cmpt 5225 × cxp 5683 dom cdm 5685 ↾ cres 5687 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 ↾f cresf 17902 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| 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-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 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-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-resf 17906 |
| This theorem is referenced by: (None) |
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