| 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 6737 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
| 3 | off2.3 | . . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) | |
| 4 | 3 | ffnd 6737 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
| 5 | off2.4 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 6 | off2.5 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 7 | eqid 2737 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵) | |
| 8 | eqidd 2738 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
| 9 | eqidd 2738 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
| 10 | 2, 4, 5, 6, 7, 8, 9 | offval 7706 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
| 11 | off2.6 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶) | |
| 12 | 11 | mpteq1d 5237 | . . 3 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
| 13 | 10, 12 | eqtrd 2777 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
| 14 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝑆) |
| 15 | inss1 4237 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
| 16 | 11, 15 | eqsstrrdi 4029 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
| 17 | 16 | sselda 3983 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴) |
| 18 | 14, 17 | ffvelcdmd 7105 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
| 19 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐺:𝐵⟶𝑇) |
| 20 | inss2 4238 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
| 21 | 11, 20 | eqsstrrdi 4029 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
| 22 | 21 | sselda 3983 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐵) |
| 23 | 19, 22 | ffvelcdmd 7105 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇) |
| 24 | off2.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) | |
| 25 | 24 | ralrimivva 3202 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
| 26 | 25 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
| 27 | ovrspc2v 7457 | . . 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 838 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
| 29 | 13, 28 | fmpt3d 7136 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 |
| This theorem is referenced by: (None) |
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