Theorem f1un 6868

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 6804	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	frnd 6744	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
3		f1f 6804	. . . 4 ⊢ (𝐺:𝐶–1-1→𝐷 → 𝐺:𝐶⟶𝐷)
4	3	frnd 6744	. . 3 ⊢ (𝐺:𝐶–1-1→𝐷 → ran 𝐺 ⊆ 𝐷)
5		unss12 4188	. . 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 6859	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
8		f1f1orn 6859	. . . . 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 771	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∩ 𝐶) = ∅)
11		ss2in 4245	. . . . . . . 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 4403	. . . . . . 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 716	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (ran 𝐹 ∩ ran 𝐺) = ∅)
16	10, 15	jca 511	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → ((𝐴 ∩ 𝐶) = ∅ ∧ (ran 𝐹 ∩ ran 𝐺) = ∅))
17		f1oun 6867	. . . 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 685	. . 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 6847	. . 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 6809	. . 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 458	. 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 685	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 395   = wceq 1540   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  ran crn 5686  –1-1→wf1 6558  –1-1-onto→wf1o 6560

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568

This theorem is referenced by:  undom  9099