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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 16552 | . . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = 〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) |
4 | 3 | fveq2d 6195 | . 2 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = (1st ‘〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉)) |
5 | fvex 6201 | . . . 4 ⊢ (1st ‘𝐹) ∈ V | |
6 | 5 | resex 5443 | . . 3 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V |
7 | dmexg 7097 | . . . 4 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V) | |
8 | mptexg 6484 | . . . 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 7180 | . . 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 695 | . 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 5990 | . . . . . 6 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆)) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
15 | 14 | dmeqd 5326 | . . . 4 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆)) |
16 | dmxpid 5345 | . . . 4 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
17 | 15, 16 | syl6eq 2672 | . . 3 ⊢ (𝜑 → dom dom 𝐻 = 𝑆) |
18 | 17 | reseq2d 5396 | . 2 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻) = ((1st ‘𝐹) ↾ 𝑆)) |
19 | 4, 11, 18 | 3eqtrd 2660 | 1 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ↾ cres 5116 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 ↾f cresf 16517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-resf 16521 |
This theorem is referenced by: (None) |
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