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Mirrors > Home > ILE Home > Th. List > funeqi | GIF version |
Description: Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
funeqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
funeqi | ⊢ (Fun 𝐴 ↔ Fun 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funeqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | funeq 5035 | . 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (Fun 𝐴 ↔ Fun 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 = wceq 1289 Fun wfun 5009 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-in 3005 df-ss 3012 df-br 3846 df-opab 3900 df-rel 4445 df-cnv 4446 df-co 4447 df-fun 5017 |
This theorem is referenced by: funmpt 5052 funmpt2 5053 funprg 5064 funtpg 5065 funtp 5067 funcnvuni 5083 f1cnvcnv 5227 f1co 5228 fun11iun 5274 f10 5287 funoprabg 5744 mpt2fun 5747 ovidig 5762 tposfun 6025 tfri1dALT 6116 tfrcl 6129 rdgfun 6138 frecfun 6160 frecfcllem 6169 th3qcor 6394 ssdomg 6493 sbthlem7 6670 sbthlemi8 6671 casefun 6774 caseinj 6778 djufun 6782 djuinj 6784 |
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