MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fco3OLD Structured version   Visualization version   GIF version

Theorem fco3OLD 6762
Description: Obsolete version of funcofd 6761 as 20-Sep-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
fco3OLD.1 (πœ‘ β†’ Fun 𝐹)
fco3OLD.2 (πœ‘ β†’ Fun 𝐺)
Assertion
Ref Expression
fco3OLD (πœ‘ β†’ (𝐹 ∘ 𝐺):(◑𝐺 β€œ dom 𝐹)⟢ran 𝐹)

Proof of Theorem fco3OLD
StepHypRef Expression
1 fco3OLD.1 . . . . 5 (πœ‘ β†’ Fun 𝐹)
2 fco3OLD.2 . . . . 5 (πœ‘ β†’ Fun 𝐺)
3 funco 6598 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺) β†’ Fun (𝐹 ∘ 𝐺))
41, 2, 3syl2anc 582 . . . 4 (πœ‘ β†’ Fun (𝐹 ∘ 𝐺))
5 fdmrn 6760 . . . 4 (Fun (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟢ran (𝐹 ∘ 𝐺))
64, 5sylib 217 . . 3 (πœ‘ β†’ (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟢ran (𝐹 ∘ 𝐺))
7 dmco 6263 . . . 4 dom (𝐹 ∘ 𝐺) = (◑𝐺 β€œ dom 𝐹)
87feq2i 6719 . . 3 ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟢ran (𝐹 ∘ 𝐺) ↔ (𝐹 ∘ 𝐺):(◑𝐺 β€œ dom 𝐹)⟢ran (𝐹 ∘ 𝐺))
96, 8sylib 217 . 2 (πœ‘ β†’ (𝐹 ∘ 𝐺):(◑𝐺 β€œ dom 𝐹)⟢ran (𝐹 ∘ 𝐺))
10 rncoss 5979 . . 3 ran (𝐹 ∘ 𝐺) βŠ† ran 𝐹
1110a1i 11 . 2 (πœ‘ β†’ ran (𝐹 ∘ 𝐺) βŠ† ran 𝐹)
129, 11fssd 6745 1 (πœ‘ β†’ (𝐹 ∘ 𝐺):(◑𝐺 β€œ dom 𝐹)⟢ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   βŠ† wss 3949  β—‘ccnv 5681  dom cdm 5682  ran crn 5683   β€œ cima 5685   ∘ ccom 5686  Fun wfun 6547  βŸΆwf 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-fun 6555  df-fn 6556  df-f 6557
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator