Theorem f1un 6736

Description: The union of two one-to-one functions with disjoint domains and codomains. (Contributed by BTernaryTau, 3-Dec-2024.)

Assertion

Ref	Expression
f1un	⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(𝐵 ∪ 𝐷))

Proof of Theorem f1un

Step	Hyp	Ref	Expression
1		f1f 6670	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	frnd 6608	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
3		f1f 6670	. . . 4 ⊢ (𝐺:𝐶–1-1→𝐷 → 𝐺:𝐶⟶𝐷)
4	3	frnd 6608	. . 3 ⊢ (𝐺:𝐶–1-1→𝐷 → ran 𝐺 ⊆ 𝐷)
5		unss12 4116	. . 3 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐺 ⊆ 𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐵 ∪ 𝐷))
6	2, 4, 5	syl2an 596	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐵 ∪ 𝐷))
7		f1f1orn 6727	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
8		f1f1orn 6727	. . . . 5 ⊢ (𝐺:𝐶–1-1→𝐷 → 𝐺:𝐶–1-1-onto→ran 𝐺)
9	7, 8	anim12i 613	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) → (𝐹:𝐴–1-1-onto→ran 𝐹 ∧ 𝐺:𝐶–1-1-onto→ran 𝐺))
10		simprl 768	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∩ 𝐶) = ∅)
11		ss2in 4170	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐺 ⊆ 𝐷) → (ran 𝐹 ∩ ran 𝐺) ⊆ (𝐵 ∩ 𝐷))
12	2, 4, 11	syl2an 596	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) → (ran 𝐹 ∩ ran 𝐺) ⊆ (𝐵 ∩ 𝐷))
13		sseq0 4333	. . . . . . 7 ⊢ (((ran 𝐹 ∩ ran 𝐺) ⊆ (𝐵 ∩ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (ran 𝐹 ∩ ran 𝐺) = ∅)
14	12, 13	sylan 580	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (ran 𝐹 ∩ ran 𝐺) = ∅)
15	14	adantrl 713	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (ran 𝐹 ∩ ran 𝐺) = ∅)
16	10, 15	jca 512	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → ((𝐴 ∩ 𝐶) = ∅ ∧ (ran 𝐹 ∩ ran 𝐺) = ∅))
17		f1oun 6735	. . . 4 ⊢ (((𝐹:𝐴–1-1-onto→ran 𝐹 ∧ 𝐺:𝐶–1-1-onto→ran 𝐺) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (ran 𝐹 ∩ ran 𝐺) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(ran 𝐹 ∪ ran 𝐺))
18	9, 16, 17	syl2an2r 682	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(ran 𝐹 ∪ ran 𝐺))
19		f1of1 6715	. . 3 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(ran 𝐹 ∪ ran 𝐺) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(ran 𝐹 ∪ ran 𝐺))
20	18, 19	syl 17	. 2 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(ran 𝐹 ∪ ran 𝐺))
21		f1ss 6676	. . 3 ⊢ (((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(ran 𝐹 ∪ ran 𝐺) ∧ (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐵 ∪ 𝐷)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(𝐵 ∪ 𝐷))
22	21	ancoms 459	. 2 ⊢ (((ran 𝐹 ∪ ran 𝐺) ⊆ (𝐵 ∪ 𝐷) ∧ (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(ran 𝐹 ∪ ran 𝐺)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(𝐵 ∪ 𝐷))
23	6, 20, 22	syl2an2r 682	1 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ran crn 5590  –1-1→wf1 6430  –1-1-onto→wf1o 6432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440

This theorem is referenced by:  undom  8846