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Theorem f1oun 5089
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-ontoB 𝐺:𝐶1-1-onto𝐷) ((A𝐶) = ∅ (B𝐷) = ∅)) → (𝐹𝐺):(A𝐶)–1-1-onto→(B𝐷))

Proof of Theorem f1oun
StepHypRef Expression
1 dff1o4 5077 . . . 4 (𝐹:A1-1-ontoB ↔ (𝐹 Fn A 𝐹 Fn B))
2 dff1o4 5077 . . . 4 (𝐺:𝐶1-1-onto𝐷 ↔ (𝐺 Fn 𝐶 𝐺 Fn 𝐷))
3 fnun 4948 . . . . . . 7 (((𝐹 Fn A 𝐺 Fn 𝐶) (A𝐶) = ∅) → (𝐹𝐺) Fn (A𝐶))
43ex 108 . . . . . 6 ((𝐹 Fn A 𝐺 Fn 𝐶) → ((A𝐶) = ∅ → (𝐹𝐺) Fn (A𝐶)))
5 fnun 4948 . . . . . . . 8 (((𝐹 Fn B 𝐺 Fn 𝐷) (B𝐷) = ∅) → (𝐹𝐺) Fn (B𝐷))
6 cnvun 4672 . . . . . . . . 9 (𝐹𝐺) = (𝐹𝐺)
76fneq1i 4936 . . . . . . . 8 ((𝐹𝐺) Fn (B𝐷) ↔ (𝐹𝐺) Fn (B𝐷))
85, 7sylibr 137 . . . . . . 7 (((𝐹 Fn B 𝐺 Fn 𝐷) (B𝐷) = ∅) → (𝐹𝐺) Fn (B𝐷))
98ex 108 . . . . . 6 ((𝐹 Fn B 𝐺 Fn 𝐷) → ((B𝐷) = ∅ → (𝐹𝐺) Fn (B𝐷)))
104, 9im2anan9 530 . . . . 5 (((𝐹 Fn A 𝐺 Fn 𝐶) (𝐹 Fn B 𝐺 Fn 𝐷)) → (((A𝐶) = ∅ (B𝐷) = ∅) → ((𝐹𝐺) Fn (A𝐶) (𝐹𝐺) Fn (B𝐷))))
1110an4s 522 . . . 4 (((𝐹 Fn A 𝐹 Fn B) (𝐺 Fn 𝐶 𝐺 Fn 𝐷)) → (((A𝐶) = ∅ (B𝐷) = ∅) → ((𝐹𝐺) Fn (A𝐶) (𝐹𝐺) Fn (B𝐷))))
121, 2, 11syl2anb 275 . . 3 ((𝐹:A1-1-ontoB 𝐺:𝐶1-1-onto𝐷) → (((A𝐶) = ∅ (B𝐷) = ∅) → ((𝐹𝐺) Fn (A𝐶) (𝐹𝐺) Fn (B𝐷))))
13 dff1o4 5077 . . 3 ((𝐹𝐺):(A𝐶)–1-1-onto→(B𝐷) ↔ ((𝐹𝐺) Fn (A𝐶) (𝐹𝐺) Fn (B𝐷)))
1412, 13syl6ibr 151 . 2 ((𝐹:A1-1-ontoB 𝐺:𝐶1-1-onto𝐷) → (((A𝐶) = ∅ (B𝐷) = ∅) → (𝐹𝐺):(A𝐶)–1-1-onto→(B𝐷)))
1514imp 115 1 (((𝐹:A1-1-ontoB 𝐺:𝐶1-1-onto𝐷) ((A𝐶) = ∅ (B𝐷) = ∅)) → (𝐹𝐺):(A𝐶)–1-1-onto→(B𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  cun 2909  cin 2910  c0 3218  ccnv 4287   Fn wfn 4840  1-1-ontowf1o 4844
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  f1oprg  5111  unen  6229
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