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Mirrors > Home > MPE Home > Th. List > fvconst | Structured version Visualization version GIF version |
Description: The value of a constant function. (Contributed by NM, 30-May-1999.) |
Ref | Expression |
---|---|
fvconst | ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffvelrn 6859 | . 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ {𝐵}) | |
2 | elsni 4533 | . 2 ⊢ ((𝐹‘𝐶) ∈ {𝐵} → (𝐹‘𝐶) = 𝐵) | |
3 | 1, 2 | syl 17 | 1 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {csn 4516 ⟶wf 6335 ‘cfv 6339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 |
This theorem is referenced by: fvconst2g 6974 fconst2g 6975 nf1const 7071 ipasslem9 28773 resf1o 30640 ccatmulgnn0dir 32091 prv1n 32964 zrtermorngc 45093 zrtermoringc 45162 |
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