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Theorem f1oun 6609
Description: The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
Assertion
Ref Expression
f1oun (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷))

Proof of Theorem f1oun
StepHypRef Expression
1 dff1o4 6598 . . . 4 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
2 dff1o4 6598 . . . 4 (𝐺:𝐶1-1-onto𝐷 ↔ (𝐺 Fn 𝐶𝐺 Fn 𝐷))
3 fnun 6434 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
43ex 416 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐶) → ((𝐴𝐶) = ∅ → (𝐹𝐺) Fn (𝐴𝐶)))
5 fnun 6434 . . . . . . . 8 (((𝐹 Fn 𝐵𝐺 Fn 𝐷) ∧ (𝐵𝐷) = ∅) → (𝐹𝐺) Fn (𝐵𝐷))
6 cnvun 5968 . . . . . . . . 9 (𝐹𝐺) = (𝐹𝐺)
76fneq1i 6420 . . . . . . . 8 ((𝐹𝐺) Fn (𝐵𝐷) ↔ (𝐹𝐺) Fn (𝐵𝐷))
85, 7sylibr 237 . . . . . . 7 (((𝐹 Fn 𝐵𝐺 Fn 𝐷) ∧ (𝐵𝐷) = ∅) → (𝐹𝐺) Fn (𝐵𝐷))
98ex 416 . . . . . 6 ((𝐹 Fn 𝐵𝐺 Fn 𝐷) → ((𝐵𝐷) = ∅ → (𝐹𝐺) Fn (𝐵𝐷)))
104, 9im2anan9 622 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐹 Fn 𝐵𝐺 Fn 𝐷)) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷))))
1110an4s 659 . . . 4 (((𝐹 Fn 𝐴𝐹 Fn 𝐵) ∧ (𝐺 Fn 𝐶𝐺 Fn 𝐷)) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷))))
121, 2, 11syl2anb 600 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷))))
13 dff1o4 6598 . . 3 ((𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷)))
1412, 13syl6ibr 255 . 2 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷)))
1514imp 410 1 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  cun 3879  cin 3880  c0 4243  ccnv 5518   Fn wfn 6319  1-1-ontowf1o 6323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331
This theorem is referenced by:  f1oprg  6634  fveqf1o  7037  oacomf1o  8174  unen  8579  enfixsn  8609  domss2  8660  isinf  8715  marypha1lem  8881  hashf1lem1  13809  f1oun2prg  14270  eupthp1  28001  isoun  30461  cycpmcl  30808  cycpmconjslem2  30847  subfacp1lem2a  32537  subfacp1lem5  32541  poimirlem3  35057  poimirlem15  35069  poimirlem16  35070  poimirlem17  35071  poimirlem19  35073  poimirlem20  35074  metakunt17  39361  eldioph2lem1  39696  eldioph2lem2  39697
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