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Theorem fncoOLD 6620
Description: Obsolete version of fnco 6619 as of 20-Sep-2024. (Contributed by NM, 22-May-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fncoOLD ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)

Proof of Theorem fncoOLD
StepHypRef Expression
1 fnfun 6603 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fnfun 6603 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
3 funco 6542 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
41, 2, 3syl2an 597 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → Fun (𝐹𝐺))
543adant3 1133 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → Fun (𝐹𝐺))
6 fndm 6606 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
76sseq2d 3977 . . . . . 6 (𝐹 Fn 𝐴 → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺𝐴))
87biimpar 479 . . . . 5 ((𝐹 Fn 𝐴 ∧ ran 𝐺𝐴) → ran 𝐺 ⊆ dom 𝐹)
9 dmcosseq 5929 . . . . 5 (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹𝐺) = dom 𝐺)
108, 9syl 17 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = dom 𝐺)
11103adant2 1132 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = dom 𝐺)
12 fndm 6606 . . . 4 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
13123ad2ant2 1135 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom 𝐺 = 𝐵)
1411, 13eqtrd 2777 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → dom (𝐹𝐺) = 𝐵)
15 df-fn 6500 . 2 ((𝐹𝐺) Fn 𝐵 ↔ (Fun (𝐹𝐺) ∧ dom (𝐹𝐺) = 𝐵))
165, 14, 15sylanbrc 584 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wss 3911  dom cdm 5634  ran crn 5635  ccom 5638  Fun wfun 6491   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500
This theorem is referenced by: (None)
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