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Theorem f1oun 6804
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 6793 . . . 4 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
2 dff1o4 6793 . . . 4 (𝐺:𝐶1-1-onto𝐷 ↔ (𝐺 Fn 𝐶𝐺 Fn 𝐷))
3 fnun 6615 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
43ex 414 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐶) → ((𝐴𝐶) = ∅ → (𝐹𝐺) Fn (𝐴𝐶)))
5 fnun 6615 . . . . . . . 8 (((𝐹 Fn 𝐵𝐺 Fn 𝐷) ∧ (𝐵𝐷) = ∅) → (𝐹𝐺) Fn (𝐵𝐷))
6 cnvun 6096 . . . . . . . . 9 (𝐹𝐺) = (𝐹𝐺)
76fneq1i 6600 . . . . . . . 8 ((𝐹𝐺) Fn (𝐵𝐷) ↔ (𝐹𝐺) Fn (𝐵𝐷))
85, 7sylibr 233 . . . . . . 7 (((𝐹 Fn 𝐵𝐺 Fn 𝐷) ∧ (𝐵𝐷) = ∅) → (𝐹𝐺) Fn (𝐵𝐷))
98ex 414 . . . . . 6 ((𝐹 Fn 𝐵𝐺 Fn 𝐷) → ((𝐵𝐷) = ∅ → (𝐹𝐺) Fn (𝐵𝐷)))
104, 9im2anan9 621 . . . . 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 599 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷))))
13 dff1o4 6793 . . 3 ((𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷)))
1412, 13syl6ibr 252 . 2 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷)))
1514imp 408 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 397   = wceq 1542  cun 3909  cin 3910  c0 4283  ccnv 5633   Fn wfn 6492  1-1-ontowf1o 6496
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by:  f1un  6805  f1oprg  6830  fveqf1o  7250  f1ofvswap  7253  oacomf1o  8513  unen  8991  enfixsn  9026  domss2  9081  isinf  9205  isinfOLD  9206  marypha1lem  9370  hashf1lem1  14354  hashf1lem1OLD  14355  f1oun2prg  14807  eupthp1  29163  isoun  31618  cycpmcl  31968  cycpmconjslem2  32007  subfacp1lem2a  33777  subfacp1lem5  33781  poimirlem3  36084  poimirlem15  36096  poimirlem16  36097  poimirlem17  36098  poimirlem19  36100  poimirlem20  36101  metakunt17  40596  eldioph2lem1  41086  eldioph2lem2  41087
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