![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > funfni | 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 5228 | . . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
2 | 1 | adantr 274 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → Fun 𝐹) |
3 | fndm 5230 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
4 | 3 | eleq2d 2210 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴)) |
5 | 4 | biimpar 295 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹) |
6 | funfni.1 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝜑) | |
7 | 2, 5, 6 | syl2anc 409 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1481 dom cdm 4547 Fun wfun 5125 Fn wfn 5126 |
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-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-cleq 2133 df-clel 2136 df-fn 5134 |
This theorem is referenced by: fneu 5235 fnbrfvb 5470 fvelrnb 5477 fvelimab 5485 fniinfv 5487 fvco2 5498 eqfnfv 5526 fndmdif 5533 fndmin 5535 elpreima 5547 fniniseg 5548 fniniseg2 5550 fnniniseg2 5551 rexsupp 5552 fnopfv 5558 fnfvelrn 5560 rexrn 5565 ralrn 5566 fsn2 5602 fnressn 5614 eufnfv 5656 rexima 5664 ralima 5665 fniunfv 5671 dff13 5677 foeqcnvco 5699 f1eqcocnv 5700 isocnv2 5721 isoini 5727 f1oiso 5735 fnovex 5812 suppssof1 6007 offveqb 6009 1stexg 6073 2ndexg 6074 smoiso 6207 rdgruledefgg 6280 rdgivallem 6286 frectfr 6305 frecrdg 6313 en1 6701 fnfi 6833 ordiso2 6928 cc2lem 7098 slotex 12025 |
Copyright terms: Public domain | W3C validator |