Theorem f1oun 6782

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 6771	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
2		dff1o4 6771	. . . 4 ⊢ (𝐺:𝐶–1-1-onto→𝐷 ↔ (𝐺 Fn 𝐶 ∧ ◡𝐺 Fn 𝐷))
3		fnun 6595	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶))
4	3	ex 412	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) → ((𝐴 ∩ 𝐶) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶)))
5		fnun 6595	. . . . . . . 8 ⊢ (((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (◡𝐹 ∪ ◡𝐺) Fn (𝐵 ∪ 𝐷))
6		cnvun 6089	. . . . . . . . 9 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
7	6	fneq1i 6578	. . . . . . . 8 ⊢ (◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷) ↔ (◡𝐹 ∪ ◡𝐺) Fn (𝐵 ∪ 𝐷))
8	5, 7	sylibr 234	. . . . . . 7 ⊢ (((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷))
9	8	ex 412	. . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷)))
10	4, 9	im2anan9 620	. . . . 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 660	. . . 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 598	. . 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 6771	. . 3 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷)))
14	12, 13	imbitrrdi 252	. 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 406	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 395   = wceq 1541   ∪ cun 3900   ∩ cin 3901  ∅c0 4283  ^◡ccnv 5615   Fn wfn 6476  –1-1-onto→wf1o 6480

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488

This theorem is referenced by:  f1un  6783  f1oprg  6808  fveqf1o  7236  f1ofvswap  7240  oacomf1o  8480  unen  8967  enfixsn  8999  domss2  9049  isinf  9149  marypha1lem  9317  hashf1lem1  14362  f1oun2prg  14824  eupthp1  30194  isoun  32681  cycpmcl  33083  cycpmconjslem2  33122  subfacp1lem2a  35222  subfacp1lem5  35226  poimirlem3  37669  poimirlem15  37681  poimirlem16  37682  poimirlem17  37683  poimirlem19  37685  poimirlem20  37686  eldioph2lem1  42799  eldioph2lem2  42800