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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcfval3 | Structured version Visualization version GIF version |
Description: General value of (𝐹 ∘f/c 𝑅𝐶) with no assumptions on functionality of 𝐹. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
Ref | Expression |
---|---|
ofcfval3 | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3416 | . . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
2 | 1 | adantr 484 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐹 ∈ V) |
3 | elex 3416 | . . 3 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ V) | |
4 | 3 | adantl 485 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 ∈ V) |
5 | dmexg 7659 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
6 | mptexg 7015 | . . . 4 ⊢ (dom 𝐹 ∈ V → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝑉 → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V) |
8 | 7 | adantr 484 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V) |
9 | simpl 486 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → 𝑓 = 𝐹) | |
10 | 9 | dmeqd 5759 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → dom 𝑓 = dom 𝐹) |
11 | 9 | fveq1d 6697 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → (𝑓‘𝑥) = (𝐹‘𝑥)) |
12 | simpr 488 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶) | |
13 | 11, 12 | oveq12d 7209 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → ((𝑓‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅𝐶)) |
14 | 10, 13 | mpteq12dv 5125 | . . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑐 = 𝐶) → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
15 | df-ofc 31730 | . . 3 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))) | |
16 | 14, 15 | ovmpoga 7341 | . 2 ⊢ ((𝐹 ∈ V ∧ 𝐶 ∈ V ∧ (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶)) ∈ V) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
17 | 2, 4, 8, 16 | syl3anc 1373 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ↦ cmpt 5120 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 ∘f/c cofc 31729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-ofc 31730 |
This theorem is referenced by: ofcfval4 31739 measdivcst 31858 |
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