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Theorem foco 6288
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 6282 . . 3 (𝐹:𝐵onto𝐶 ↔ (𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶))
2 dffo2 6282 . . 3 (𝐺:𝐴onto𝐵 ↔ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵))
3 fco 6220 . . . . 5 ((𝐹:𝐵𝐶𝐺:𝐴𝐵) → (𝐹𝐺):𝐴𝐶)
43ad2ant2r 800 . . . 4 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → (𝐹𝐺):𝐴𝐶)
5 fdm 6213 . . . . . . . 8 (𝐹:𝐵𝐶 → dom 𝐹 = 𝐵)
6 eqtr3 2782 . . . . . . . 8 ((dom 𝐹 = 𝐵 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
75, 6sylan 489 . . . . . . 7 ((𝐹:𝐵𝐶 ∧ ran 𝐺 = 𝐵) → dom 𝐹 = ran 𝐺)
8 rncoeq 5545 . . . . . . . . 9 (dom 𝐹 = ran 𝐺 → ran (𝐹𝐺) = ran 𝐹)
98eqeq1d 2763 . . . . . . . 8 (dom 𝐹 = ran 𝐺 → (ran (𝐹𝐺) = 𝐶 ↔ ran 𝐹 = 𝐶))
109biimpar 503 . . . . . . 7 ((dom 𝐹 = ran 𝐺 ∧ ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
117, 10sylan 489 . . . . . 6 (((𝐹:𝐵𝐶 ∧ ran 𝐺 = 𝐵) ∧ ran 𝐹 = 𝐶) → ran (𝐹𝐺) = 𝐶)
1211an32s 881 . . . . 5 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ ran 𝐺 = 𝐵) → ran (𝐹𝐺) = 𝐶)
1312adantrl 754 . . . 4 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → ran (𝐹𝐺) = 𝐶)
144, 13jca 555 . . 3 (((𝐹:𝐵𝐶 ∧ ran 𝐹 = 𝐶) ∧ (𝐺:𝐴𝐵 ∧ ran 𝐺 = 𝐵)) → ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
151, 2, 14syl2anb 497 . 2 ((𝐹:𝐵onto𝐶𝐺:𝐴onto𝐵) → ((𝐹𝐺):𝐴𝐶 ∧ ran (𝐹𝐺) = 𝐶))
16 dffo2 6282 . 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 383   = wceq 1632  dom cdm 5267  ran crn 5268  ccom 5271  wf 6046  ontowfo 6048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-fun 6052  df-fn 6053  df-f 6054  df-fo 6056
This theorem is referenced by:  f1oco  6322  wdomtr  8648  fin1a2lem7  9441  cofull  16816  uniiccdif  23567
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