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Mirrors > Home > ILE Home > Th. List > fconst2g | GIF version |
Description: A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.) |
Ref | Expression |
---|---|
fconst2g | ⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvconst 5351 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | |
2 | 1 | adantlr 446 | . . . . . 6 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
3 | fvconst2g 5375 | . . . . . . 7 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵) | |
4 | 3 | adantll 445 | . . . . . 6 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵) |
5 | 2, 4 | eqtr4d 2075 | . . . . 5 ⊢ (((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥)) |
6 | 5 | ralrimiva 2392 | . . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥)) |
7 | ffn 5046 | . . . . 5 ⊢ (𝐹:𝐴⟶{𝐵} → 𝐹 Fn 𝐴) | |
8 | fnconstg 5084 | . . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) Fn 𝐴) | |
9 | eqfnfv 5265 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 × {𝐵}) Fn 𝐴) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥))) | |
10 | 7, 8, 9 | syl2an 273 | . . . 4 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → (𝐹 = (𝐴 × {𝐵}) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = ((𝐴 × {𝐵})‘𝑥))) |
11 | 6, 10 | mpbird 156 | . . 3 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐵 ∈ 𝐶) → 𝐹 = (𝐴 × {𝐵})) |
12 | 11 | expcom 109 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} → 𝐹 = (𝐴 × {𝐵}))) |
13 | fconstg 5083 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | |
14 | feq1 5030 | . . 3 ⊢ (𝐹 = (𝐴 × {𝐵}) → (𝐹:𝐴⟶{𝐵} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) | |
15 | 13, 14 | syl5ibrcom 146 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐹 = (𝐴 × {𝐵}) → 𝐹:𝐴⟶{𝐵})) |
16 | 12, 15 | impbid 120 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∀wral 2306 {csn 3375 × cxp 4343 Fn wfn 4897 ⟶wf 4898 ‘cfv 4902 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 |
This theorem is referenced by: fconst2 5378 |
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