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Mirrors > Home > ILE Home > Th. List > fvconst2g | GIF version |
Description: The value of a constant function. (Contributed by NM, 20-Aug-2005.) |
Ref | Expression |
---|---|
fvconst2g | ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconstg 5394 | . 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | |
2 | fvconst 5684 | . 2 ⊢ (((𝐴 × {𝐵}):𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵) | |
3 | 1, 2 | sylan 281 | 1 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 {csn 3583 × cxp 4609 ⟶wf 5194 ‘cfv 5198 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 |
This theorem is referenced by: fconst2g 5711 fvconst2 5712 ser0 10470 exp3vallem 10477 exp3val 10478 exp1 10482 expp1 10483 resqrexlem1arp 10969 resqrexlemf1 10972 climconst2 11254 climaddc1 11292 climmulc2 11294 climsubc1 11295 climsubc2 11296 climlec2 11304 prodf1 11505 prod0 11548 ialgrlemconst 11997 ialgr0 11998 algrf 11999 algrp1 12000 0mhm 12704 lmconst 13010 cnconst2 13027 dvidlemap 13454 dvconst 13455 dvef 13482 |
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