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Mirrors > Home > MPE Home > Th. List > funfni | Structured version Visualization version GIF version |
Description: Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.) |
Ref | Expression |
---|---|
funfni.1 | ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝜑) |
Ref | Expression |
---|---|
funfni | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 6446 | . 2 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
2 | fndm 6448 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | 2 | eleq2d 2895 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴)) |
4 | 3 | biimpar 478 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹) |
5 | funfni.1 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝜑) | |
6 | 1, 4, 5 | syl2an2r 681 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 dom cdm 5548 Fun wfun 6342 Fn wfn 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-clel 2890 df-fn 6351 |
This theorem is referenced by: fneu 6454 elpreima 6820 fnopfv 6835 fnfvelrn 6840 funressnfv 43155 fnafvelrn 43245 afvco2 43252 fnafv2elrn 43309 fnbrafv2b 43324 |
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