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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 2165 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
7 | eqidd 2165 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ofrfval 6052 | . . . . 5 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
9 | 8 | biimpa 294 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥)) |
10 | fveq2 5480 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
11 | fveq2 5480 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋)) | |
12 | 10, 11 | breq12d 3989 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ (𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
13 | 12 | rspccv 2822 | . . . 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 1189 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋)𝑅(𝐺‘𝑋)) |
16 | simp1 986 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝜑) | |
17 | inss1 3337 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
18 | 5, 17 | eqsstrri 3170 | . . . 4 ⊢ 𝑆 ⊆ 𝐴 |
19 | simp3 988 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
20 | 18, 19 | sseldi 3135 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐴) |
21 | ofrval.6 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) | |
22 | 16, 20, 21 | syl2anc 409 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶) |
23 | inss2 3338 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
24 | 5, 23 | eqsstrri 3170 | . . . 4 ⊢ 𝑆 ⊆ 𝐵 |
25 | 24, 19 | sseldi 3135 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵) |
26 | ofrval.7 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) | |
27 | 16, 25, 26 | syl2anc 409 | . 2 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷) |
28 | 15, 22, 27 | 3brtr3d 4007 | 1 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 ∀wral 2442 ∩ cin 3110 class class class wbr 3976 Fn wfn 5177 ‘cfv 5182 ∘𝑟 cofr 6043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ofr 6045 |
This theorem is referenced by: (None) |
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