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Theorem f1oun 5273
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 5261 . . . 4 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
2 dff1o4 5261 . . . 4 (𝐺:𝐶1-1-onto𝐷 ↔ (𝐺 Fn 𝐶𝐺 Fn 𝐷))
3 fnun 5120 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
43ex 113 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐶) → ((𝐴𝐶) = ∅ → (𝐹𝐺) Fn (𝐴𝐶)))
5 fnun 5120 . . . . . . . 8 (((𝐹 Fn 𝐵𝐺 Fn 𝐷) ∧ (𝐵𝐷) = ∅) → (𝐹𝐺) Fn (𝐵𝐷))
6 cnvun 4837 . . . . . . . . 9 (𝐹𝐺) = (𝐹𝐺)
76fneq1i 5108 . . . . . . . 8 ((𝐹𝐺) Fn (𝐵𝐷) ↔ (𝐹𝐺) Fn (𝐵𝐷))
85, 7sylibr 132 . . . . . . 7 (((𝐹 Fn 𝐵𝐺 Fn 𝐷) ∧ (𝐵𝐷) = ∅) → (𝐹𝐺) Fn (𝐵𝐷))
98ex 113 . . . . . 6 ((𝐹 Fn 𝐵𝐺 Fn 𝐷) → ((𝐵𝐷) = ∅ → (𝐹𝐺) Fn (𝐵𝐷)))
104, 9im2anan9 565 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐹 Fn 𝐵𝐺 Fn 𝐷)) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷))))
1110an4s 555 . . . 4 (((𝐹 Fn 𝐴𝐹 Fn 𝐵) ∧ (𝐺 Fn 𝐶𝐺 Fn 𝐷)) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷))))
121, 2, 11syl2anb 285 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷))))
13 dff1o4 5261 . . 3 ((𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷)))
1412, 13syl6ibr 160 . 2 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷)))
1514imp 122 1 (((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  cun 2997  cin 2998  c0 3286  ccnv 4437   Fn wfn 5010  1-1-ontowf1o 5014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022
This theorem is referenced by:  f1oprg  5295  unen  6531  zfz1isolem1  10241
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