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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 16599 | . . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = 〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) |
4 | 3 | fveq2d 6233 | . 2 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = (1st ‘〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉)) |
5 | fvex 6239 | . . . 4 ⊢ (1st ‘𝐹) ∈ V | |
6 | 5 | resex 5478 | . . 3 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V |
7 | dmexg 7139 | . . . 4 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V) | |
8 | mptexg 6525 | . . . 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 7222 | . . 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 696 | . 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 | fndm 6028 | . . . . . 6 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆)) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
15 | 14 | dmeqd 5358 | . . . 4 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆)) |
16 | dmxpid 5377 | . . . 4 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
17 | 15, 16 | syl6eq 2701 | . . 3 ⊢ (𝜑 → dom dom 𝐻 = 𝑆) |
18 | 17 | reseq2d 5428 | . 2 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻) = ((1st ‘𝐹) ↾ 𝑆)) |
19 | 4, 11, 18 | 3eqtrd 2689 | 1 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 Vcvv 3231 〈cop 4216 ↦ cmpt 4762 × cxp 5141 dom cdm 5143 ↾ cres 5145 Fn wfn 5921 ‘cfv 5926 (class class class)co 6690 1st c1st 7208 2nd c2nd 7209 ↾f cresf 16564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-resf 16568 |
This theorem is referenced by: (None) |
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