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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 6655 | . 2 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
2 | fndm 6658 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | 2 | eleq2d 2811 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴)) |
4 | 3 | biimpar 476 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹) |
5 | funfni.1 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝜑) | |
6 | 1, 4, 5 | syl2an2r 683 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 dom cdm 5678 Fun wfun 6543 Fn wfn 6544 |
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-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-cleq 2717 df-clel 2802 df-fn 6552 |
This theorem is referenced by: fneu 6665 elpreima 7066 fnopfv 7084 fnfvelrn 7089 funressnfv 46563 fnafvelrn 46687 afvco2 46694 fnafv2elrn 46751 fnbrafv2b 46766 |
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