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Theorem foco 6594
 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 6586 . . 3 (𝐹:𝐵onto𝐶 ↔ (𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶))
2 dffo2 6586 . . 3 (𝐺:𝐴onto𝐵 ↔ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵))
3 fco 6522 . . . . 5 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
43ad2ant2r 746 . . . 4 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → (𝐹𝐺):𝐴𝐶)
5 fdm 6512 . . . . . . . 8 (𝐹:𝐵𝐶 → dom 𝐹 = 𝐵)
6 eqtr3 2781 . . . . . . . 8 ((dom 𝐹 = 𝐵 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
75, 6sylan 583 . . . . . . 7 ((𝐹:𝐵𝐶 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
8 rncoeq 5822 . . . . . . . . 9 (dom 𝐹 = ran 𝐺 → ran (𝐹𝐺) = ran 𝐹)
98eqeq1d 2761 . . . . . . . 8 (dom 𝐹 = ran 𝐺 → (ran (𝐹𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
109biimpar 481 . . . . . . 7 ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
117, 10sylan 583 . . . . . 6 (((𝐹:𝐵𝐶 ∧ ran 𝐺 = 𝐵) ∧ ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
1211an32s 651 . . . . 5 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = 𝐵) → ran (𝐹𝐺) = 𝐶)
1312adantrl 715 . . . 4 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → ran (𝐹𝐺) = 𝐶)
144, 13jca 515 . . 3 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
151, 2, 14syl2anb 600 . 2 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
16 dffo2 6586 . 2 ((𝐹𝐺):𝐴onto𝐶 ↔ ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
1715, 16sylibr 237 1 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1539  dom cdm 5529  ran crn 5530   ∘ ccom 5533  ⟶wf 6337  –onto→wfo 6339 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5038  df-opab 5100  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-fun 6343  df-fn 6344  df-f 6345  df-fo 6347 This theorem is referenced by:  f1oco  6630  wdomtr  9086  fin1a2lem7  9880  cofull  17278  sursubmefmnd  18142  uniiccdif  24293
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