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Mirrors > Home > ILE Home > Th. List > ofrval | 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 |
ofrval.6 | ⊢ ((φ ∧ 𝑋 ∈ A) → (𝐹‘𝑋) = 𝐶) |
ofrval.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 2038 | . . . . . 6 ⊢ ((φ ∧ x ∈ A) → (𝐹‘x) = (𝐹‘x)) | |
7 | eqidd 2038 | . . . . . 6 ⊢ ((φ ∧ x ∈ B) → (𝐺‘x) = (𝐺‘x)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ofrfval 5662 | . . . . 5 ⊢ (φ → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀x ∈ 𝑆 (𝐹‘x)𝑅(𝐺‘x))) |
9 | 8 | biimpa 280 | . . . 4 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺) → ∀x ∈ 𝑆 (𝐹‘x)𝑅(𝐺‘x)) |
10 | fveq2 5121 | . . . . . 6 ⊢ (x = 𝑋 → (𝐹‘x) = (𝐹‘𝑋)) | |
11 | fveq2 5121 | . . . . . 6 ⊢ (x = 𝑋 → (𝐺‘x) = (𝐺‘𝑋)) | |
12 | 10, 11 | breq12d 3768 | . . . . 5 ⊢ (x = 𝑋 → ((𝐹‘x)𝑅(𝐺‘x) ↔ (𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
13 | 12 | rspccv 2647 | . . . 4 ⊢ (∀x ∈ 𝑆 (𝐹‘x)𝑅(𝐺‘x) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
14 | 9, 13 | syl 14 | . . 3 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
15 | 14 | 3impia 1100 | . 2 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋)𝑅(𝐺‘𝑋)) |
16 | simp1 903 | . . 3 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → φ) | |
17 | inss1 3151 | . . . . 5 ⊢ (A ∩ B) ⊆ A | |
18 | 5, 17 | eqsstr3i 2970 | . . . 4 ⊢ 𝑆 ⊆ A |
19 | simp3 905 | . . . 4 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
20 | 18, 19 | sseldi 2937 | . . 3 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ A) |
21 | ofrval.6 | . . 3 ⊢ ((φ ∧ 𝑋 ∈ A) → (𝐹‘𝑋) = 𝐶) | |
22 | 16, 20, 21 | syl2anc 391 | . 2 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶) |
23 | inss2 3152 | . . . . 5 ⊢ (A ∩ B) ⊆ B | |
24 | 5, 23 | eqsstr3i 2970 | . . . 4 ⊢ 𝑆 ⊆ B |
25 | 24, 19 | sseldi 2937 | . . 3 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ B) |
26 | ofrval.7 | . . 3 ⊢ ((φ ∧ 𝑋 ∈ B) → (𝐺‘𝑋) = 𝐷) | |
27 | 16, 25, 26 | syl2anc 391 | . 2 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷) |
28 | 15, 22, 27 | 3brtr3d 3784 | 1 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 ∀wral 2300 ∩ cin 2910 class class class wbr 3755 Fn wfn 4840 ‘cfv 4845 ∘𝑟 cofr 5653 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ofr 5655 |
This theorem is referenced by: (None) |
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