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Theorem foun 6848
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 6804 . . . 4 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 fofn 6804 . . . 4 (𝐺:𝐶onto𝐷𝐺 Fn 𝐶)
31, 2anim12i 614 . . 3 ((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) → (𝐹 Fn 𝐴𝐺 Fn 𝐶))
4 fnun 6660 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
53, 4sylan 581 . 2 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
6 rnun 6142 . . 3 ran (𝐹𝐺) = (ran 𝐹 ∪ ran 𝐺)
7 forn 6805 . . . . 5 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87ad2antrr 725 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran 𝐹 = 𝐵)
9 forn 6805 . . . . 5 (𝐺:𝐶onto𝐷 → ran 𝐺 = 𝐷)
109ad2antlr 726 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran 𝐺 = 𝐷)
118, 10uneq12d 4163 . . 3 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (ran 𝐹 ∪ ran 𝐺) = (𝐵𝐷))
126, 11eqtrid 2785 . 2 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran (𝐹𝐺) = (𝐵𝐷))
13 df-fo 6546 . 2 ((𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐶) ∧ ran (𝐹𝐺) = (𝐵𝐷)))
145, 12, 13sylanbrc 584 1 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  cun 3945  cin 3946  c0 4321  ran crn 5676   Fn wfn 6535  ontowfo 6538
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546
This theorem is referenced by: (None)
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