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Theorem f1oun 5146
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 5134 . . . 4 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
2 dff1o4 5134 . . . 4 (𝐺:𝐶1-1-onto𝐷 ↔ (𝐺 Fn 𝐶𝐺 Fn 𝐷))
3 fnun 5005 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺) Fn (𝐴𝐶))
43ex 108 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐶) → ((𝐴𝐶) = ∅ → (𝐹𝐺) Fn (𝐴𝐶)))
5 fnun 5005 . . . . . . . 8 (((𝐹 Fn 𝐵𝐺 Fn 𝐷) ∧ (𝐵𝐷) = ∅) → (𝐹𝐺) Fn (𝐵𝐷))
6 cnvun 4729 . . . . . . . . 9 (𝐹𝐺) = (𝐹𝐺)
76fneq1i 4993 . . . . . . . 8 ((𝐹𝐺) Fn (𝐵𝐷) ↔ (𝐹𝐺) Fn (𝐵𝐷))
85, 7sylibr 137 . . . . . . 7 (((𝐹 Fn 𝐵𝐺 Fn 𝐷) ∧ (𝐵𝐷) = ∅) → (𝐹𝐺) Fn (𝐵𝐷))
98ex 108 . . . . . 6 ((𝐹 Fn 𝐵𝐺 Fn 𝐷) → ((𝐵𝐷) = ∅ → (𝐹𝐺) Fn (𝐵𝐷)))
104, 9im2anan9 530 . . . . 5 (((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐹 Fn 𝐵𝐺 Fn 𝐷)) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷))))
1110an4s 522 . . . 4 (((𝐹 Fn 𝐴𝐹 Fn 𝐵) ∧ (𝐺 Fn 𝐶𝐺 Fn 𝐷)) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷))))
121, 2, 11syl2anb 275 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷))))
13 dff1o4 5134 . . 3 ((𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷) ↔ ((𝐹𝐺) Fn (𝐴𝐶) ∧ (𝐹𝐺) Fn (𝐵𝐷)))
1412, 13syl6ibr 151 . 2 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) → (((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷)))
1514imp 115 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 97   = wceq 1243  cun 2915  cin 2916  c0 3224  ccnv 4344   Fn wfn 4897  1-1-ontowf1o 4901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909
This theorem is referenced by:  f1oprg  5168  unen  6293
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