![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 17943 | . . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = 〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) |
4 | 3 | fveq2d 6911 | . 2 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = (1st ‘〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉)) |
5 | fvex 6920 | . . . 4 ⊢ (1st ‘𝐹) ∈ V | |
6 | 5 | resex 6049 | . . 3 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V |
7 | dmexg 7924 | . . . 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 8025 | . . 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 6674 | . . . . 5 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
14 | 13 | dmeqd 5919 | . . . 4 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆)) |
15 | dmxpid 5944 | . . . 4 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
16 | 14, 15 | eqtrdi 2791 | . . 3 ⊢ (𝜑 → dom dom 𝐻 = 𝑆) |
17 | 16 | reseq2d 6000 | . 2 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻) = ((1st ‘𝐹) ↾ 𝑆)) |
18 | 4, 11, 17 | 3eqtrd 2779 | 1 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ↦ cmpt 5231 × cxp 5687 dom cdm 5689 ↾ cres 5691 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 ↾f cresf 17908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-resf 17912 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |