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Mirrors > Home > MPE Home > Th. List > fococnv2 | Structured version Visualization version GIF version |
Description: The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
Ref | Expression |
---|---|
fococnv2 | ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofun 6590 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) | |
2 | funcocnv2 6638 | . . 3 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
4 | forn 6592 | . . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) | |
5 | 4 | reseq2d 5852 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵)) |
6 | 3, 5 | eqtrd 2856 | 1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 I cid 5458 ◡ccnv 5553 ran crn 5555 ↾ cres 5556 ∘ ccom 5558 Fun wfun 6348 –onto→wfo 6352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-fun 6356 df-fn 6357 df-f 6358 df-fo 6360 |
This theorem is referenced by: f1ococnv2 6640 foeqcnvco 7055 |
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