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| Mirrors > Home > MPE Home > Th. List > ofrval | Structured version Visualization version GIF version | ||
| Description: Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| Ref | Expression |
|---|---|
| offval.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
| offval.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
| offval.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| offval.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| offval.5 | ⊢ (𝐴 ∩ 𝐵) = 𝑆 |
| ofval.6 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) |
| ofval.7 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) |
| Ref | Expression |
|---|---|
| ofrval | ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | offval.1 | . . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
| 2 | offval.2 | . . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
| 3 | offval.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 4 | offval.4 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 5 | offval.5 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
| 6 | eqidd 2738 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
| 7 | eqidd 2738 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | ofrfval 7707 | . . . . 5 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
| 9 | 8 | biimpa 476 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥)) |
| 10 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
| 11 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋)) | |
| 12 | 10, 11 | breq12d 5156 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ (𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
| 13 | 12 | rspccv 3619 | . . . 4 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
| 14 | 9, 13 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
| 15 | 14 | 3impia 1118 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋)𝑅(𝐺‘𝑋)) |
| 16 | simp1 1137 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝜑) | |
| 17 | inss1 4237 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
| 18 | 5, 17 | eqsstrri 4031 | . . . 4 ⊢ 𝑆 ⊆ 𝐴 |
| 19 | simp3 1139 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
| 20 | 18, 19 | sselid 3981 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐴) |
| 21 | ofval.6 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) | |
| 22 | 16, 20, 21 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶) |
| 23 | inss2 4238 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
| 24 | 5, 23 | eqsstrri 4031 | . . . 4 ⊢ 𝑆 ⊆ 𝐵 |
| 25 | 24, 19 | sselid 3981 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵) |
| 26 | ofval.7 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) | |
| 27 | 16, 25, 26 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷) |
| 28 | 15, 22, 27 | 3brtr3d 5174 | 1 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 class class class wbr 5143 Fn wfn 6556 ‘cfv 6561 ∘r cofr 7696 |
| 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-ofr 7698 |
| This theorem is referenced by: mhpmulcl 22153 itg1le 25748 gsumle 33101 ftc1anclem5 37704 |
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