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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:𝐴1-1𝐵𝐺:𝐶1-1𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1→(𝐵𝐷))

Proof of Theorem f1un
StepHypRef Expression
1 f1f 6804 . . . 4 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
21frnd 6744 . . 3 (𝐹:𝐴1-1𝐵 → ran 𝐹𝐵)
3 f1f 6804 . . . 4 (𝐺:𝐶1-1𝐷𝐺:𝐶𝐷)
43frnd 6744 . . 3 (𝐺:𝐶1-1𝐷 → ran 𝐺𝐷)
5 unss12 4188 . . 3 ((ran 𝐹𝐵 ∧ ran 𝐺𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐵𝐷))
62, 4, 5syl2an 596 . 2 ((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐵𝐷))
7 f1f1orn 6859 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
8 f1f1orn 6859 . . . . 5 (𝐺:𝐶1-1𝐷𝐺:𝐶1-1-onto→ran 𝐺)
97, 8anim12i 613 . . . 4 ((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) → (𝐹:𝐴1-1-onto→ran 𝐹𝐺:𝐶1-1-onto→ran 𝐺))
10 simprl 771 . . . . 5 (((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐴𝐶) = ∅)
11 ss2in 4245 . . . . . . . 8 ((ran 𝐹𝐵 ∧ ran 𝐺𝐷) → (ran 𝐹 ∩ ran 𝐺) ⊆ (𝐵𝐷))
122, 4, 11syl2an 596 . . . . . . 7 ((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) → (ran 𝐹 ∩ ran 𝐺) ⊆ (𝐵𝐷))
13 sseq0 4403 . . . . . . 7 (((ran 𝐹 ∩ ran 𝐺) ⊆ (𝐵𝐷) ∧ (𝐵𝐷) = ∅) → (ran 𝐹 ∩ ran 𝐺) = ∅)
1412, 13sylan 580 . . . . . 6 (((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) ∧ (𝐵𝐷) = ∅) → (ran 𝐹 ∩ ran 𝐺) = ∅)
1514adantrl 716 . . . . 5 (((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (ran 𝐹 ∩ ran 𝐺) = ∅)
1610, 15jca 511 . . . 4 (((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → ((𝐴𝐶) = ∅ ∧ (ran 𝐹 ∩ ran 𝐺) = ∅))
17 f1oun 6867 . . . 4 (((𝐹:𝐴1-1-onto→ran 𝐹𝐺:𝐶1-1-onto→ran 𝐺) ∧ ((𝐴𝐶) = ∅ ∧ (ran 𝐹 ∩ ran 𝐺) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(ran 𝐹 ∪ ran 𝐺))
189, 16, 17syl2an2r 685 . . 3 (((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(ran 𝐹 ∪ ran 𝐺))
19 f1of1 6847 . . 3 ((𝐹𝐺):(𝐴𝐶)–1-1-onto→(ran 𝐹 ∪ ran 𝐺) → (𝐹𝐺):(𝐴𝐶)–1-1→(ran 𝐹 ∪ ran 𝐺))
2018, 19syl 17 . 2 (((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1→(ran 𝐹 ∪ ran 𝐺))
21 f1ss 6809 . . 3 (((𝐹𝐺):(𝐴𝐶)–1-1→(ran 𝐹 ∪ ran 𝐺) ∧ (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐵𝐷)) → (𝐹𝐺):(𝐴𝐶)–1-1→(𝐵𝐷))
2221ancoms 458 . 2 (((ran 𝐹 ∪ ran 𝐺) ⊆ (𝐵𝐷) ∧ (𝐹𝐺):(𝐴𝐶)–1-1→(ran 𝐹 ∪ ran 𝐺)) → (𝐹𝐺):(𝐴𝐶)–1-1→(𝐵𝐷))
236, 20, 22syl2an2r 685 1 (((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1→(𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  cun 3949  cin 3950  wss 3951  c0 4333  ran crn 5686  1-1wf1 6558  1-1-ontowf1o 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
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