Theorem f1oun 6821

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 6810	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
2		dff1o4 6810	. . . 4 ⊢ (𝐺:𝐶–1-1-onto→𝐷 ↔ (𝐺 Fn 𝐶 ∧ ◡𝐺 Fn 𝐷))
3		fnun 6630	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶))
4	3	ex 416	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) → ((𝐴 ∩ 𝐶) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶)))
5		fnun 6630	. . . . . . . 8 ⊢ (((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (◡𝐹 ∪ ◡𝐺) Fn (𝐵 ∪ 𝐷))
6		cnvun 6122	. . . . . . . . 9 ⊢ ◡(𝐹 ∪ 𝐺) = (◡𝐹 ∪ ◡𝐺)
7	6	fneq1i 6613	. . . . . . . 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 416	. . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ ◡𝐺 Fn 𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷)))
10	4, 9	im2anan9 629	. . . . 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 670	. . . 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 607	. . 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 6810	. . 3 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐶) ∧ ◡(𝐹 ∪ 𝐺) Fn (𝐵 ∪ 𝐷)))
14	12, 13	imbitrrdi 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 410	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 399   = wceq 1559   ∪ cun 3900   ∩ cin 3901  ∅c0 4283  ^◡ccnv 5642   Fn wfn 6511  –1-1-onto→wf1o 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523

This theorem is referenced by:  f1un  6822  f1oprg  6848  fveqf1o  7281  f1ofvswap  7285  oacomf1o  8528  unen  9020  enfixsn  9052  domss2  9102  isinf  9203  marypha1lem  9373  hashf1lem1  14462  f1oun2prg  14924  eupthp1  30375  isoun  32865  cycpmcl  33257  cycpmconjslem2  33296  subfacp1lem2a  35491  subfacp1lem5  35495  poimirlem3  38083  poimirlem15  38095  poimirlem16  38096  poimirlem17  38097  poimirlem19  38099  poimirlem20  38100  eldioph2lem1  43302  eldioph2lem2  43303