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Theorem foco 5144
Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
foco ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)

Proof of Theorem foco
StepHypRef Expression
1 dffo2 5138 . . 3 (𝐹:𝐵onto𝐶 ↔ (𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶))
2 dffo2 5138 . . 3 (𝐺:𝐴onto𝐵 ↔ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵))
3 fco 5084 . . . . 5 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
43ad2ant2r 486 . . . 4 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → (𝐹𝐺):𝐴𝐶)
5 fdm 5078 . . . . . . . 8 (𝐹:𝐵𝐶 → dom 𝐹 = 𝐵)
6 eqtr3 2075 . . . . . . . 8 ((dom 𝐹 = 𝐵 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
75, 6sylan 271 . . . . . . 7 ((𝐹:𝐵𝐶 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
8 rncoeq 4633 . . . . . . . . 9 (dom 𝐹 = ran 𝐺 → ran (𝐹𝐺) = ran 𝐹)
98eqeq1d 2064 . . . . . . . 8 (dom 𝐹 = ran 𝐺 → (ran (𝐹𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
109biimpar 285 . . . . . . 7 ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
117, 10sylan 271 . . . . . 6 (((𝐹:𝐵𝐶 ∧ ran 𝐺 = 𝐵) ∧ ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
1211an32s 510 . . . . 5 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = 𝐵) → ran (𝐹𝐺) = 𝐶)
1312adantrl 455 . . . 4 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → ran (𝐹𝐺) = 𝐶)
144, 13jca 294 . . 3 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
151, 2, 14syl2anb 279 . 2 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
16 dffo2 5138 . 2 ((𝐹𝐺):𝐴onto𝐶 ↔ ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
1715, 16sylibr 141 1 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  dom cdm 4373  ran crn 4374  ccom 4377  wf 4926  ontowfo 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936
This theorem is referenced by:  f1oco  5177
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