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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 5048 | . 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 104 = wceq 1290 Fun wfun 5022 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-in 3006 df-ss 3013 df-br 3852 df-opab 3906 df-rel 4459 df-cnv 4460 df-co 4461 df-fun 5030 |
This theorem is referenced by: funmpt 5065 funmpt2 5066 funprg 5077 funtpg 5078 funtp 5080 funcnvuni 5096 f1cnvcnv 5240 f1co 5241 fun11iun 5287 f10 5300 funoprabg 5758 mpt2fun 5761 ovidig 5776 tposfun 6039 tfri1dALT 6130 tfrcl 6143 rdgfun 6152 frecfun 6174 frecfcllem 6183 th3qcor 6410 ssdomg 6549 sbthlem7 6726 sbthlemi8 6727 casefun 6830 caseinj 6834 djufun 6840 djuinj 6842 strleund 11643 strleun 11644 1strbas 11654 2strbasg 11656 2stropg 11657 |
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