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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 16375 | . . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = 〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉) |
4 | 3 | fveq2d 6107 | . 2 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = (1st ‘〈((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))〉)) |
5 | fvex 6113 | . . . 4 ⊢ (1st ‘𝐹) ∈ V | |
6 | 5 | resex 5363 | . . 3 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V |
7 | dmexg 6989 | . . . 4 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V) | |
8 | mptexg 6389 | . . . 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 7071 | . . 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 694 | . 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 5904 | . . . . . 6 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆)) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
15 | 14 | dmeqd 5248 | . . . 4 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆)) |
16 | dmxpid 5266 | . . . 4 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
17 | 15, 16 | syl6eq 2660 | . . 3 ⊢ (𝜑 → dom dom 𝐻 = 𝑆) |
18 | 17 | reseq2d 5317 | . 2 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻) = ((1st ‘𝐹) ↾ 𝑆)) |
19 | 4, 11, 18 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ↦ cmpt 4643 × cxp 5036 dom cdm 5038 ↾ cres 5040 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 ↾f cresf 16340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-resf 16344 |
This theorem is referenced by: (None) |
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