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Theorem f1un 6869
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 6805 . . . 4 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
21frnd 6745 . . 3 (𝐹:𝐴1-1𝐵 → ran 𝐹𝐵)
3 f1f 6805 . . . 4 (𝐺:𝐶1-1𝐷𝐺:𝐶𝐷)
43frnd 6745 . . 3 (𝐺:𝐶1-1𝐷 → ran 𝐺𝐷)
5 unss12 4198 . . 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 6860 . . . . 5 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
8 f1f1orn 6860 . . . . 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 4253 . . . . . . . 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 4409 . . . . . . 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 6868 . . . 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 6848 . . 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 6810 . . 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 1537  cun 3961  cin 3962  wss 3963  c0 4339  ran crn 5690  1-1wf1 6560  1-1-ontowf1o 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  undom  9098
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