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Theorem fco3OLD 6744
Description: Obsolete version of funcofd 6743 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 6581 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺) β†’ Fun (𝐹 ∘ 𝐺))
41, 2, 3syl2anc 583 . . . 4 (πœ‘ β†’ Fun (𝐹 ∘ 𝐺))
5 fdmrn 6742 . . . 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 6246 . . . 4 dom (𝐹 ∘ 𝐺) = (◑𝐺 β€œ dom 𝐹)
87feq2i 6702 . . 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 5964 . . 3 ran (𝐹 ∘ 𝐺) βŠ† ran 𝐹
1110a1i 11 . 2 (πœ‘ β†’ ran (𝐹 ∘ 𝐺) βŠ† ran 𝐹)
129, 11fssd 6728 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 3943  β—‘ccnv 5668  dom cdm 5669  ran crn 5670   β€œ cima 5672   ∘ ccom 5673  Fun wfun 6530  βŸΆwf 6532
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fun 6538  df-fn 6539  df-f 6540
This theorem is referenced by: (None)
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