Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > off2 | Structured version Visualization version GIF version |
Description: The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.) |
Ref | Expression |
---|---|
off2.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) |
off2.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
off2.3 | ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) |
off2.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
off2.5 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
off2.6 | ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶) |
Ref | Expression |
---|---|
off2 | ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | off2.2 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
2 | 1 | ffnd 6515 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | off2.3 | . . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) | |
4 | 3 | ffnd 6515 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
5 | off2.4 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | off2.5 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
7 | eqid 2821 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵) | |
8 | eqidd 2822 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
9 | eqidd 2822 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
10 | 2, 4, 5, 6, 7, 8, 9 | offval 7416 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
11 | off2.6 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶) | |
12 | 11 | mpteq1d 5155 | . . 3 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
13 | 10, 12 | eqtrd 2856 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
14 | 1 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝑆) |
15 | inss1 4205 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
16 | 11, 15 | eqsstrrdi 4022 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
17 | 16 | sselda 3967 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴) |
18 | 14, 17 | ffvelrnd 6852 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
19 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐺:𝐵⟶𝑇) |
20 | inss2 4206 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
21 | 11, 20 | eqsstrrdi 4022 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
22 | 21 | sselda 3967 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐵) |
23 | 19, 22 | ffvelrnd 6852 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇) |
24 | off2.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) | |
25 | 24 | ralrimivva 3191 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
26 | 25 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
27 | ovrspc2v 7182 | . . 3 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ (𝐺‘𝑧) ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) | |
28 | 18, 23, 26, 27 | syl21anc 835 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
29 | 13, 28 | fmpt3d 6880 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∩ cin 3935 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 |
This theorem is referenced by: (None) |
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