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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 5200 | . . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | 1 | simprbi 273 | . 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I ) |
3 | iss 4937 | . . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹))) | |
4 | dfdm4 4803 | . . . . . . . 8 ⊢ dom 𝐹 = ran ◡𝐹 | |
5 | dmcoeq 4883 | . . . . . . . 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 4622 | . . . . . . 7 ⊢ ran 𝐹 = dom ◡𝐹 | |
8 | 6, 7 | eqtr4i 2194 | . . . . . 6 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹 |
9 | 8 | a1i 9 | . . . . 5 ⊢ (Fun 𝐹 → dom (𝐹 ∘ ◡𝐹) = ran 𝐹) |
10 | 9 | reseq2d 4891 | . . . 4 ⊢ (Fun 𝐹 → ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹)) |
11 | 10 | eqeq2d 2182 | . . 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 1348 ⊆ wss 3121 I cid 4273 ◡ccnv 4610 dom cdm 4611 ran crn 4612 ↾ cres 4613 ∘ ccom 4615 Rel wrel 4616 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-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-fun 5200 |
This theorem is referenced by: fococnv2 5468 f1cocnv2 5470 funcoeqres 5473 |
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