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Theorem f1un 6850
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→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (𝐡 ∩ 𝐷) = βˆ…)) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1β†’(𝐡 βˆͺ 𝐷))
Proof of Theorem f1un
StepHypRef Expression
1 f1f 6784 . . . 4 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴⟢𝐡)
21frnd 6722 . . 3 (𝐹:𝐴–1-1→𝐡 β†’ ran 𝐹 βŠ† 𝐡)
3 f1f 6784 . . . 4 (𝐺:𝐢–1-1→𝐷 β†’ 𝐺:𝐢⟢𝐷)
43frnd 6722 . . 3 (𝐺:𝐢–1-1→𝐷 β†’ ran 𝐺 βŠ† 𝐷)
5 unss12 4181 . . 3 ((ran 𝐹 βŠ† 𝐡 ∧ ran 𝐺 βŠ† 𝐷) β†’ (ran 𝐹 βˆͺ ran 𝐺) βŠ† (𝐡 βˆͺ 𝐷))
62, 4, 5syl2an 596 . 2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) β†’ (ran 𝐹 βˆͺ ran 𝐺) βŠ† (𝐡 βˆͺ 𝐷))
7 f1f1orn 6841 . . . . 5 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
8 f1f1orn 6841 . . . . 5 (𝐺:𝐢–1-1→𝐷 β†’ 𝐺:𝐢–1-1-ontoβ†’ran 𝐺)
97, 8anim12i 613 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) β†’ (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ∧ 𝐺:𝐢–1-1-ontoβ†’ran 𝐺))
10 simprl 769 . . . . 5 (((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (𝐡 ∩ 𝐷) = βˆ…)) β†’ (𝐴 ∩ 𝐢) = βˆ…)
11 ss2in 4235 . . . . . . . 8 ((ran 𝐹 βŠ† 𝐡 ∧ ran 𝐺 βŠ† 𝐷) β†’ (ran 𝐹 ∩ ran 𝐺) βŠ† (𝐡 ∩ 𝐷))
122, 4, 11syl2an 596 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) β†’ (ran 𝐹 ∩ ran 𝐺) βŠ† (𝐡 ∩ 𝐷))
13 sseq0 4398 . . . . . . 7 (((ran 𝐹 ∩ ran 𝐺) βŠ† (𝐡 ∩ 𝐷) ∧ (𝐡 ∩ 𝐷) = βˆ…) β†’ (ran 𝐹 ∩ ran 𝐺) = βˆ…)
1412, 13sylan 580 . . . . . 6 (((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ (𝐡 ∩ 𝐷) = βˆ…) β†’ (ran 𝐹 ∩ ran 𝐺) = βˆ…)
1514adantrl 714 . . . . 5 (((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (𝐡 ∩ 𝐷) = βˆ…)) β†’ (ran 𝐹 ∩ ran 𝐺) = βˆ…)
1610, 15jca 512 . . . 4 (((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (𝐡 ∩ 𝐷) = βˆ…)) β†’ ((𝐴 ∩ 𝐢) = βˆ… ∧ (ran 𝐹 ∩ ran 𝐺) = βˆ…))
17 f1oun 6849 . . . 4 (((𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ∧ 𝐺:𝐢–1-1-ontoβ†’ran 𝐺) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (ran 𝐹 ∩ ran 𝐺) = βˆ…)) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1-ontoβ†’(ran 𝐹 βˆͺ ran 𝐺))
189, 16, 17syl2an2r 683 . . 3 (((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (𝐡 ∩ 𝐷) = βˆ…)) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1-ontoβ†’(ran 𝐹 βˆͺ ran 𝐺))
19 f1of1 6829 . . 3 ((𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1-ontoβ†’(ran 𝐹 βˆͺ ran 𝐺) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1β†’(ran 𝐹 βˆͺ ran 𝐺))
2018, 19syl 17 . 2 (((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (𝐡 ∩ 𝐷) = βˆ…)) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1β†’(ran 𝐹 βˆͺ ran 𝐺))
21 f1ss 6790 . . 3 (((𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1β†’(ran 𝐹 βˆͺ ran 𝐺) ∧ (ran 𝐹 βˆͺ ran 𝐺) βŠ† (𝐡 βˆͺ 𝐷)) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1β†’(𝐡 βˆͺ 𝐷))
2221ancoms 459 . 2 (((ran 𝐹 βˆͺ ran 𝐺) βŠ† (𝐡 βˆͺ 𝐷) ∧ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1β†’(ran 𝐹 βˆͺ ran 𝐺)) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1β†’(𝐡 βˆͺ 𝐷))
236, 20, 22syl2an2r 683 1 (((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (𝐡 ∩ 𝐷) = βˆ…)) β†’ (𝐹 βˆͺ 𝐺):(𝐴 βˆͺ 𝐢)–1-1β†’(𝐡 βˆͺ 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  ran crn 5676  β€“1-1β†’wf1 6537  β€“1-1-ontoβ†’wf1o 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547
This theorem is referenced by:  undom  9055
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