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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 5190 | . 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 1335 Fun wfun 5164 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-in 3108 df-ss 3115 df-br 3966 df-opab 4026 df-rel 4593 df-cnv 4594 df-co 4595 df-fun 5172 |
This theorem is referenced by: funopg 5204 funsng 5216 funcnvuni 5239 f1eq1 5370 frecuzrdgtclt 10320 shftfn 10724 ennnfonelemfun 12146 ennnfonelemf1 12147 isstruct2im 12200 isstruct2r 12201 structfung 12207 setsfun 12225 setsfun0 12226 funmptd 13378 |
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