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Mirrors > Home > ILE Home > Th. List > off | GIF version |
Description: The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
off.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) |
off.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
off.3 | ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) |
off.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
off.5 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
off.6 | ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
Ref | Expression |
---|---|
off | ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | off.2 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
2 | off.6 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) = 𝐶 | |
3 | inss1 3367 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
4 | 2, 3 | eqsstrri 3200 | . . . . . 6 ⊢ 𝐶 ⊆ 𝐴 |
5 | 4 | sseli 3163 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴) |
6 | ffvelcdm 5662 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝑆 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆) | |
7 | 1, 5, 6 | syl2an 289 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
8 | off.3 | . . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) | |
9 | inss2 3368 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
10 | 2, 9 | eqsstrri 3200 | . . . . . 6 ⊢ 𝐶 ⊆ 𝐵 |
11 | 10 | sseli 3163 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵) |
12 | ffvelcdm 5662 | . . . . 5 ⊢ ((𝐺:𝐵⟶𝑇 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) ∈ 𝑇) | |
13 | 8, 11, 12 | syl2an 289 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇) |
14 | off.1 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) | |
15 | 14 | ralrimivva 2569 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
16 | 15 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
17 | oveq1 5895 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥𝑅𝑦) = ((𝐹‘𝑧)𝑅𝑦)) | |
18 | 17 | eleq1d 2256 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈)) |
19 | oveq2 5896 | . . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑧)𝑅𝑦) = ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) | |
20 | 19 | eleq1d 2256 | . . . . 5 ⊢ (𝑦 = (𝐺‘𝑧) → (((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)) |
21 | 18, 20 | rspc2va 2867 | . . . 4 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ (𝐺‘𝑧) ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
22 | 7, 13, 16, 21 | syl21anc 1247 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
23 | eqid 2187 | . . 3 ⊢ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) | |
24 | 22, 23 | fmptd 5683 | . 2 ⊢ (𝜑 → (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈) |
25 | ffn 5377 | . . . . 5 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴) | |
26 | 1, 25 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
27 | ffn 5377 | . . . . 5 ⊢ (𝐺:𝐵⟶𝑇 → 𝐺 Fn 𝐵) | |
28 | 8, 27 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
29 | off.4 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
30 | off.5 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
31 | eqidd 2188 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
32 | eqidd 2188 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
33 | 26, 28, 29, 30, 2, 31, 32 | offval 6104 | . . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
34 | 33 | feq1d 5364 | . 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈 ↔ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈)) |
35 | 24, 34 | mpbird 167 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 ∀wral 2465 ∩ cin 3140 ↦ cmpt 4076 Fn wfn 5223 ⟶wf 5224 ‘cfv 5228 (class class class)co 5888 ∘𝑓 cof 6095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-of 6097 |
This theorem is referenced by: offeq 6110 lcomf 13573 dvaddxxbr 14518 dvmulxxbr 14519 dvaddxx 14520 dvmulxx 14521 dviaddf 14522 dvimulf 14523 |
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