Theorem f1oun 6819

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 6808	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
2		dff1o4 6808	. . . 4 ⊢ (𝐺:𝐶–1-1-onto→𝐷 ↔ (𝐺 Fn 𝐶 ∧ ◡𝐺 Fn 𝐷))
3		fnun 6632	. . . . . . 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 6632	. . . . . . . 8 ⊢ (((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (◡𝐹 ∪ ◡𝐺) Fn (𝐵 ∪ 𝐷))
6		cnvun 6115	. . . . . . . . 9 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
7	6	fneq1i 6615	. . . . . . . 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 6808	. . 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 1540   ∪ cun 3912   ∩ cin 3913  ∅c0 4296  ^◡ccnv 5637   Fn wfn 6506  –1-1-onto→wf1o 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518

This theorem is referenced by:  f1un  6820  f1oprg  6845  fveqf1o  7277  f1ofvswap  7281  oacomf1o  8529  unen  9017  enfixsn  9050  domss2  9100  isinf  9207  isinfOLD  9208  marypha1lem  9384  hashf1lem1  14420  f1oun2prg  14883  eupthp1  30145  isoun  32625  cycpmcl  33073  cycpmconjslem2  33112  subfacp1lem2a  35167  subfacp1lem5  35171  poimirlem3  37617  poimirlem15  37629  poimirlem16  37630  poimirlem17  37631  poimirlem19  37633  poimirlem20  37634  eldioph2lem1  42748  eldioph2lem2  42749