Theorem f1oun 6634

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:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷))

Proof of Theorem f1oun

Step	Hyp	Ref	Expression
1		dff1o4 6623	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
2		dff1o4 6623	. . . 4 ⊢ (𝐺:𝐶–1-1-onto→𝐷 ↔ (𝐺 Fn 𝐶 ∧ ◡𝐺 Fn 𝐷))
3		fnun 6463	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶))
4	3	ex 415	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) → ((𝐴 ∩ 𝐶) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶)))
5		fnun 6463	. . . . . . . 8 ⊢ (((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (◡𝐹 ∪ ◡𝐺) Fn (𝐵 ∪ 𝐷))
6		cnvun 6001	. . . . . . . . 9 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
7	6	fneq1i 6450	. . . . . . . 8 ⊢ (◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷) ↔ (◡𝐹 ∪ ◡𝐺) Fn (𝐵 ∪ 𝐷))
8	5, 7	sylibr 236	. . . . . . 7 ⊢ (((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷))
9	8	ex 415	. . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷)))
10	4, 9	im2anan9 621	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷)) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷))))
11	10	an4s 658	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵) ∧ (𝐺 Fn 𝐶 ∧ ◡𝐺 Fn 𝐷)) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷))))
12	1, 2, 11	syl2anb 599	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷))))
13		dff1o4 6623	. . 3 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷)))
14	12, 13	syl6ibr 254	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷)))
15	14	imp 409	1 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∪ cun 3934   ∩ cin 3935  ∅c0 4291  ^◡ccnv 5554   Fn wfn 6350  –1-1-onto→wf1o 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362

This theorem is referenced by:  f1oprg  6659  fveqf1o  7058  oacomf1o  8191  unen  8596  enfixsn  8626  domss2  8676  isinf  8731  marypha1lem  8897  hashf1lem1  13814  f1oun2prg  14279  eupthp1  27995  isoun  30437  cycpmcl  30758  cycpmconjslem2  30797  subfacp1lem2a  32427  subfacp1lem5  32431  poimirlem3  34910  poimirlem15  34922  poimirlem16  34923  poimirlem17  34924  poimirlem19  34926  poimirlem20  34927  eldioph2lem1  39377  eldioph2lem2  39378