Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > funcocnv2 | GIF version |
Description: Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
funcocnv2 | ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fun 4924 | . . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | 1 | simprbi 269 | . 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I ) |
3 | iss 4674 | . . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹))) | |
4 | dfdm4 4545 | . . . . . . . 8 ⊢ dom 𝐹 = ran ◡𝐹 | |
5 | dmcoeq 4622 | . . . . . . . 8 ⊢ (dom 𝐹 = ran ◡𝐹 → dom (𝐹 ∘ ◡𝐹) = dom ◡𝐹) | |
6 | 4, 5 | ax-mp 7 | . . . . . . 7 ⊢ dom (𝐹 ∘ ◡𝐹) = dom ◡𝐹 |
7 | df-rn 4374 | . . . . . . 7 ⊢ ran 𝐹 = dom ◡𝐹 | |
8 | 6, 7 | eqtr4i 2104 | . . . . . 6 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹 |
9 | 8 | a1i 9 | . . . . 5 ⊢ (Fun 𝐹 → dom (𝐹 ∘ ◡𝐹) = ran 𝐹) |
10 | 9 | reseq2d 4630 | . . . 4 ⊢ (Fun 𝐹 → ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹)) |
11 | 10 | eqeq2d 2092 | . . 3 ⊢ (Fun 𝐹 → ((𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)) ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))) |
12 | 3, 11 | syl5bb 190 | . 2 ⊢ (Fun 𝐹 → ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))) |
13 | 2, 12 | mpbid 145 | 1 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ⊆ wss 2973 I cid 4043 ◡ccnv 4362 dom cdm 4363 ran crn 4364 ↾ cres 4365 ∘ ccom 4367 Rel wrel 4368 Fun wfun 4916 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-fun 4924 |
This theorem is referenced by: fococnv2 5172 f1cocnv2 5174 funcoeqres 5177 |
Copyright terms: Public domain | W3C validator |