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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 5218 | . 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 1348 Fun wfun 5192 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-in 3127 df-ss 3134 df-br 3990 df-opab 4051 df-rel 4618 df-cnv 4619 df-co 4620 df-fun 5200 |
This theorem is referenced by: funopg 5232 funsng 5244 funcnvuni 5267 f1eq1 5398 frecuzrdgtclt 10377 shftfn 10788 ennnfonelemfun 12372 ennnfonelemf1 12373 isstruct2im 12426 isstruct2r 12427 structfung 12433 setsfun 12451 setsfun0 12452 funmptd 13838 |
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