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Mirrors > Home > ILE Home > Th. List > fconstg | GIF version |
Description: A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.) |
Ref | Expression |
---|---|
fconstg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3571 | . . . 4 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵}) | |
2 | 1 | xpeq2d 4610 | . . 3 ⊢ (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵})) |
3 | feq1 5302 | . . . 4 ⊢ ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥})) | |
4 | feq3 5304 | . . . 4 ⊢ ({𝑥} = {𝐵} → ((𝐴 × {𝐵}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) | |
5 | 3, 4 | sylan9bb 458 | . . 3 ⊢ (((𝐴 × {𝑥}) = (𝐴 × {𝐵}) ∧ {𝑥} = {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) |
6 | 2, 1, 5 | syl2anc 409 | . 2 ⊢ (𝑥 = 𝐵 → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) |
7 | vex 2715 | . . 3 ⊢ 𝑥 ∈ V | |
8 | 7 | fconst 5365 | . 2 ⊢ (𝐴 × {𝑥}):𝐴⟶{𝑥} |
9 | 6, 8 | vtoclg 2772 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1335 ∈ wcel 2128 {csn 3560 × cxp 4584 ⟶wf 5166 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-fun 5172 df-fn 5173 df-f 5174 |
This theorem is referenced by: fnconstg 5367 fconst6g 5368 xpsng 5642 fvconst2g 5681 fconst2g 5682 dvef 13088 |
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