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Theorem foco 5059
Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
foco ((𝐹:Bonto𝐶 𝐺:AontoB) → (𝐹𝐺):Aonto𝐶)

Proof of Theorem foco
StepHypRef Expression
1 dffo2 5053 . . 3 (𝐹:Bonto𝐶 ↔ (𝐹:B𝐶 ran 𝐹 = 𝐶))
2 dffo2 5053 . . 3 (𝐺:AontoB ↔ (𝐺:AB ran 𝐺 = B))
3 fco 4999 . . . . 5 ((𝐹:B𝐶 𝐺:AB) → (𝐹𝐺):A𝐶)
43ad2ant2r 478 . . . 4 (((𝐹:B𝐶 ran 𝐹 = 𝐶) (𝐺:AB ran 𝐺 = B)) → (𝐹𝐺):A𝐶)
5 fdm 4993 . . . . . . . 8 (𝐹:B𝐶 → dom 𝐹 = B)
6 eqtr3 2056 . . . . . . . 8 ((dom 𝐹 = B ran 𝐺 = B) → dom 𝐹 = ran 𝐺)
75, 6sylan 267 . . . . . . 7 ((𝐹:B𝐶 ran 𝐺 = B) → dom 𝐹 = ran 𝐺)
8 rncoeq 4548 . . . . . . . . 9 (dom 𝐹 = ran 𝐺 → ran (𝐹𝐺) = ran 𝐹)
98eqeq1d 2045 . . . . . . . 8 (dom 𝐹 = ran 𝐺 → (ran (𝐹𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
109biimpar 281 . . . . . . 7 ((dom 𝐹 = ran 𝐺 ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
117, 10sylan 267 . . . . . 6 (((𝐹:B𝐶 ran 𝐺 = B) ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
1211an32s 502 . . . . 5 (((𝐹:B𝐶 ran 𝐹 = 𝐶) ran 𝐺 = B) → ran (𝐹𝐺) = 𝐶)
1312adantrl 447 . . . 4 (((𝐹:B𝐶 ran 𝐹 = 𝐶) (𝐺:AB ran 𝐺 = B)) → ran (𝐹𝐺) = 𝐶)
144, 13jca 290 . . 3 (((𝐹:B𝐶 ran 𝐹 = 𝐶) (𝐺:AB ran 𝐺 = B)) → ((𝐹𝐺):A𝐶 ran (𝐹𝐺) = 𝐶))
151, 2, 14syl2anb 275 . 2 ((𝐹:Bonto𝐶 𝐺:AontoB) → ((𝐹𝐺):A𝐶 ran (𝐹𝐺) = 𝐶))
16 dffo2 5053 . 2 ((𝐹𝐺):Aonto𝐶 ↔ ((𝐹𝐺):A𝐶 ran (𝐹𝐺) = 𝐶))
1715, 16sylibr 137 1 ((𝐹:Bonto𝐶 𝐺:AontoB) → (𝐹𝐺):Aonto𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  dom cdm 4288  ran crn 4289  ccom 4292  wf 4841  ontowfo 4843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851
This theorem is referenced by:  f1oco  5092
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