Theorem f1un 6882

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 6817	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2	1	frnd 6755	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
3		f1f 6817	. . . 4 ⊢ (𝐺:𝐶–1-1→𝐷 → 𝐺:𝐶⟶𝐷)
4	3	frnd 6755	. . 3 ⊢ (𝐺:𝐶–1-1→𝐷 → ran 𝐺 ⊆ 𝐷)
5		unss12 4211	. . 3 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐺 ⊆ 𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐵 ∪ 𝐷))
6	2, 4, 5	syl2an 595	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐵 ∪ 𝐷))
7		f1f1orn 6873	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
8		f1f1orn 6873	. . . . 5 ⊢ (𝐺:𝐶–1-1→𝐷 → 𝐺:𝐶–1-1-onto→ran 𝐺)
9	7, 8	anim12i 612	. . . 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 770	. . . . 5 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∩ 𝐶) = ∅)
11		ss2in 4266	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ ran 𝐺 ⊆ 𝐷) → (ran 𝐹 ∩ ran 𝐺) ⊆ (𝐵 ∩ 𝐷))
12	2, 4, 11	syl2an 595	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) → (ran 𝐹 ∩ ran 𝐺) ⊆ (𝐵 ∩ 𝐷))
13		sseq0 4426	. . . . . . 7 ⊢ (((ran 𝐹 ∩ ran 𝐺) ⊆ (𝐵 ∩ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (ran 𝐹 ∩ ran 𝐺) = ∅)
14	12, 13	sylan 579	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (ran 𝐹 ∩ ran 𝐺) = ∅)
15	14	adantrl 715	. . . . 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 6881	. . . 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 684	. . 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 6861	. . 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 6822	. . 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 684	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 1537   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ran crn 5701  –1-1→wf1 6570  –1-1-onto→wf1o 6572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580

This theorem is referenced by:  undom  9125