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Theorem foco 6084
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 6078 . . 3 (𝐹:𝐵onto𝐶 ↔ (𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶))
2 dffo2 6078 . . 3 (𝐺:𝐴onto𝐵 ↔ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵))
3 fco 6017 . . . . 5 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
43ad2ant2r 782 . . . 4 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → (𝐹𝐺):𝐴𝐶)
5 fdm 6010 . . . . . . . 8 (𝐹:𝐵𝐶 → dom 𝐹 = 𝐵)
6 eqtr3 2647 . . . . . . . 8 ((dom 𝐹 = 𝐵 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
75, 6sylan 488 . . . . . . 7 ((𝐹:𝐵𝐶 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
8 rncoeq 5353 . . . . . . . . 9 (dom 𝐹 = ran 𝐺 → ran (𝐹𝐺) = ran 𝐹)
98eqeq1d 2628 . . . . . . . 8 (dom 𝐹 = ran 𝐺 → (ran (𝐹𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
109biimpar 502 . . . . . . 7 ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
117, 10sylan 488 . . . . . 6 (((𝐹:𝐵𝐶 ∧ ran 𝐺 = 𝐵) ∧ ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
1211an32s 845 . . . . 5 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = 𝐵) → ran (𝐹𝐺) = 𝐶)
1312adantrl 751 . . . 4 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → ran (𝐹𝐺) = 𝐶)
144, 13jca 554 . . 3 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
151, 2, 14syl2anb 496 . 2 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
16 dffo2 6078 . 2 ((𝐹𝐺):𝐴onto𝐶 ↔ ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
1715, 16sylibr 224 1 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → (𝐹𝐺):𝐴onto𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  dom cdm 5079  ran crn 5080  ccom 5083  wf 5846  ontowfo 5848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-fun 5852  df-fn 5853  df-f 5854  df-fo 5856
This theorem is referenced by:  f1oco  6118  wdomtr  8425  fin1a2lem7  9173  cofull  16510  uniiccdif  23247
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