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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 3603 | . . . 4 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵}) | |
2 | 1 | xpeq2d 4650 | . . 3 ⊢ (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵})) |
3 | feq1 5348 | . . . 4 ⊢ ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥})) | |
4 | feq3 5350 | . . . 4 ⊢ ({𝑥} = {𝐵} → ((𝐴 × {𝐵}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) | |
5 | 3, 4 | sylan9bb 462 | . . 3 ⊢ (((𝐴 × {𝑥}) = (𝐴 × {𝐵}) ∧ {𝑥} = {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) |
6 | 2, 1, 5 | syl2anc 411 | . 2 ⊢ (𝑥 = 𝐵 → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) |
7 | vex 2740 | . . 3 ⊢ 𝑥 ∈ V | |
8 | 7 | fconst 5411 | . 2 ⊢ (𝐴 × {𝑥}):𝐴⟶{𝑥} |
9 | 6, 8 | vtoclg 2797 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 {csn 3592 × cxp 4624 ⟶wf 5212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-fun 5218 df-fn 5219 df-f 5220 |
This theorem is referenced by: fnconstg 5413 fconst6g 5414 xpsng 5691 fvconst2g 5730 fconst2g 5731 dvef 14081 |
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