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Mirrors > Home > ILE Home > Th. List > funeqd | GIF version |
Description: Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
Ref | Expression |
---|---|
funeqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
funeqd | ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funeqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | funeq 5138 | . 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 Fun wfun 5112 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-in 3072 df-ss 3079 df-br 3925 df-opab 3985 df-rel 4541 df-cnv 4542 df-co 4543 df-fun 5120 |
This theorem is referenced by: funopg 5152 funsng 5164 funcnvuni 5187 f1eq1 5318 frecuzrdgtclt 10187 shftfn 10589 ennnfonelemfun 11919 ennnfonelemf1 11920 isstruct2im 11958 isstruct2r 11959 structfung 11965 setsfun 11983 setsfun0 11984 |
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