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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 5193 | . 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Fun 𝐴 ↔ Fun 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1335 Fun wfun 5167 |
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 3968 df-opab 4029 df-rel 4596 df-cnv 4597 df-co 4598 df-fun 5175 |
This theorem is referenced by: funmpt 5211 funmpt2 5212 funprg 5223 funtpg 5224 funtp 5226 funcnvuni 5242 f1cnvcnv 5389 f1co 5390 fun11iun 5438 f10 5451 funoprabg 5923 mpofun 5926 ovidig 5941 tposfun 6210 tfri1dALT 6301 tfrcl 6314 rdgfun 6323 frecfun 6345 frecfcllem 6354 th3qcor 6587 ssdomg 6726 sbthlem7 6910 sbthlemi8 6911 casefun 7032 caseinj 7036 djufun 7051 djuinj 7053 ctssdccl 7058 axaddf 7791 axmulf 7792 strleund 12374 strleun 12375 1strbas 12385 2strbasg 12387 2stropg 12388 |
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