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Theorem foun 5172
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 5135 . . . 4 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 fofn 5135 . . . 4 (𝐺:𝐶onto𝐷𝐺 Fn 𝐶)
31, 2anim12i 325 . . 3 ((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) → (𝐹 Fn 𝐴𝐺 Fn 𝐶))
4 fnun 5032 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
53, 4sylan 271 . 2 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
6 rnun 4759 . . 3 ran (𝐹𝐺) = (ran 𝐹 ∪ ran 𝐺)
7 forn 5136 . . . . 5 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87ad2antrr 465 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran 𝐹 = 𝐵)
9 forn 5136 . . . . 5 (𝐺:𝐶onto𝐷 → ran 𝐺 = 𝐷)
109ad2antlr 466 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran 𝐺 = 𝐷)
118, 10uneq12d 3125 . . 3 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (ran 𝐹 ∪ ran 𝐺) = (𝐵𝐷))
126, 11syl5eq 2100 . 2 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran (𝐹𝐺) = (𝐵𝐷))
13 df-fo 4935 . 2 ((𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐶) ∧ ran (𝐹𝐺) = (𝐵𝐷)))
145, 12, 13sylanbrc 402 1 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  cun 2942  cin 2943  c0 3251  ran crn 4373   Fn wfn 4924  ontowfo 4927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-id 4057  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935
This theorem is referenced by: (None)
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