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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 5218 | . 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 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: funmpt 5236 funmpt2 5237 funprg 5248 funtpg 5249 funtp 5251 funcnvuni 5267 f1cnvcnv 5414 f1co 5415 fun11iun 5463 f10 5476 funoprabg 5952 mpofun 5955 ovidig 5970 tposfun 6239 tfri1dALT 6330 tfrcl 6343 rdgfun 6352 frecfun 6374 frecfcllem 6383 th3qcor 6617 ssdomg 6756 sbthlem7 6940 sbthlemi8 6941 casefun 7062 caseinj 7066 djufun 7081 djuinj 7083 ctssdccl 7088 axaddf 7830 axmulf 7831 strleund 12506 strleun 12507 1strbas 12517 2strbasg 12519 2stropg 12520 |
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