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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 5173 | . . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | 1 | simprbi 273 | . 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I ) |
3 | iss 4913 | . . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹))) | |
4 | dfdm4 4779 | . . . . . . . 8 ⊢ dom 𝐹 = ran ◡𝐹 | |
5 | dmcoeq 4859 | . . . . . . . 8 ⊢ (dom 𝐹 = ran ◡𝐹 → dom (𝐹 ∘ ◡𝐹) = dom ◡𝐹) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ dom (𝐹 ∘ ◡𝐹) = dom ◡𝐹 |
7 | df-rn 4598 | . . . . . . 7 ⊢ ran 𝐹 = dom ◡𝐹 | |
8 | 6, 7 | eqtr4i 2181 | . . . . . 6 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹 |
9 | 8 | a1i 9 | . . . . 5 ⊢ (Fun 𝐹 → dom (𝐹 ∘ ◡𝐹) = ran 𝐹) |
10 | 9 | reseq2d 4867 | . . . 4 ⊢ (Fun 𝐹 → ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹)) |
11 | 10 | eqeq2d 2169 | . . 3 ⊢ (Fun 𝐹 → ((𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)) ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))) |
12 | 3, 11 | syl5bb 191 | . 2 ⊢ (Fun 𝐹 → ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))) |
13 | 2, 12 | mpbid 146 | 1 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ⊆ wss 3102 I cid 4249 ◡ccnv 4586 dom cdm 4587 ran crn 4588 ↾ cres 4589 ∘ ccom 4591 Rel wrel 4592 Fun wfun 5165 |
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-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-fun 5173 |
This theorem is referenced by: fococnv2 5441 f1cocnv2 5443 funcoeqres 5446 |
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