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Mirrors > Home > ILE Home > Th. List > fvconst2 | GIF version |
Description: The value of a constant function. (Contributed by NM, 16-Apr-2005.) |
Ref | Expression |
---|---|
fvconst2.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fvconst2 | ⊢ (𝐶 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvconst2.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | fvconst2g 5407 | . 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵) | |
3 | 1, 2 | mpan 415 | 1 ⊢ (𝐶 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 Vcvv 2602 {csn 3406 × cxp 4369 ‘cfv 4932 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ral 2354 df-rex 2355 df-v 2604 df-sbc 2817 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-mpt 3849 df-id 4056 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-fv 4940 |
This theorem is referenced by: ovconst2 5683 iser0f 9569 iserige0 10319 |
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