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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 | ffvelcdm 7073 | . 2 ⊢ ((𝐹:𝐴⟶{𝐵} ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ {𝐵}) | |
2 | elsni 4637 | . 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 395 = wceq 1533 ∈ wcel 2098 {csn 4620 ⟶wf 6529 ‘cfv 6533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 |
This theorem is referenced by: fvconst2g 7195 fconst2g 7196 f1cdmsn 7272 nf1const 7294 zrtermorngc 20528 zrtermoringc 20560 ipasslem9 30526 resf1o 32390 ccatmulgnn0dir 34008 prv1n 34877 sticksstones11 41431 |
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