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Theorem foun 5432
 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 5393 . . . 4 (𝐹:𝐴onto𝐵𝐹 Fn 𝐴)
2 fofn 5393 . . . 4 (𝐺:𝐶onto𝐷𝐺 Fn 𝐶)
31, 2anim12i 336 . . 3 ((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) → (𝐹 Fn 𝐴𝐺 Fn 𝐶))
4 fnun 5275 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
53, 4sylan 281 . 2 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
6 rnun 4993 . . 3 ran (𝐹𝐺) = (ran 𝐹 ∪ ran 𝐺)
7 forn 5394 . . . . 5 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
87ad2antrr 480 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran 𝐹 = 𝐵)
9 forn 5394 . . . . 5 (𝐺:𝐶onto𝐷 → ran 𝐺 = 𝐷)
109ad2antlr 481 . . . 4 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran 𝐺 = 𝐷)
118, 10uneq12d 3262 . . 3 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (ran 𝐹 ∪ ran 𝐺) = (𝐵𝐷))
126, 11syl5eq 2202 . 2 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → ran (𝐹𝐺) = (𝐵𝐷))
13 df-fo 5175 . 2 ((𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐶) ∧ ran (𝐹𝐺) = (𝐵𝐷)))
145, 12, 13sylanbrc 414 1 (((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1335   ∪ cun 3100   ∩ cin 3101  ∅c0 3394  ran crn 4586   Fn wfn 5164  –onto→wfo 5167 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-id 4253  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-fun 5171  df-fn 5172  df-f 5173  df-fo 5175 This theorem is referenced by: (None)
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