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Theorem foun 6866
Description: The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
foun (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷))

Proof of Theorem foun
StepHypRef Expression
1 fofn 6822 . . . 4 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 fofn 6822 . . . 4 (𝐺:𝐶onto𝐷𝐺 Fn 𝐶)
31, 2anim12i 613 . . 3 ((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) → (𝐹 Fn 𝐴𝐺 Fn 𝐶))
4 fnun 6682 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
53, 4sylan 580 . 2 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
6 rnun 6165 . . 3 ran (𝐹𝐺) = (ran 𝐹 ∪ ran 𝐺)
7 forn 6823 . . . . 5 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87ad2antrr 726 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran 𝐹 = 𝐵)
9 forn 6823 . . . . 5 (𝐺:𝐶onto𝐷 → ran 𝐺 = 𝐷)
109ad2antlr 727 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran 𝐺 = 𝐷)
118, 10uneq12d 4169 . . 3 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (ran 𝐹 ∪ ran 𝐺) = (𝐵𝐷))
126, 11eqtrid 2789 . 2 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran (𝐹𝐺) = (𝐵𝐷))
13 df-fo 6567 . 2 ((𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐶) ∧ ran (𝐹𝐺) = (𝐵𝐷)))
145, 12, 13sylanbrc 583 1 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  cun 3949  cin 3950  c0 4333  ran crn 5686   Fn wfn 6556  ontowfo 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567
This theorem is referenced by: (None)
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