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Theorem f1un 6805
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 6739 . . . 4 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴⟢𝐡)
21frnd 6677 . . 3 (𝐹:𝐴–1-1→𝐡 β†’ ran 𝐹 βŠ† 𝐡)
3 f1f 6739 . . . 4 (𝐺:𝐢–1-1→𝐷 β†’ 𝐺:𝐢⟢𝐷)
43frnd 6677 . . 3 (𝐺:𝐢–1-1→𝐷 β†’ ran 𝐺 βŠ† 𝐷)
5 unss12 4143 . . 3 ((ran 𝐹 βŠ† 𝐡 ∧ ran 𝐺 βŠ† 𝐷) β†’ (ran 𝐹 βˆͺ ran 𝐺) βŠ† (𝐡 βˆͺ 𝐷))
62, 4, 5syl2an 597 . 2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) β†’ (ran 𝐹 βˆͺ ran 𝐺) βŠ† (𝐡 βˆͺ 𝐷))
7 f1f1orn 6796 . . . . 5 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
8 f1f1orn 6796 . . . . 5 (𝐺:𝐢–1-1→𝐷 β†’ 𝐺:𝐢–1-1-ontoβ†’ran 𝐺)
97, 8anim12i 614 . . . 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 770 . . . . 5 (((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ ((𝐴 ∩ 𝐢) = βˆ… ∧ (𝐡 ∩ 𝐷) = βˆ…)) β†’ (𝐴 ∩ 𝐢) = βˆ…)
11 ss2in 4197 . . . . . . . 8 ((ran 𝐹 βŠ† 𝐡 ∧ ran 𝐺 βŠ† 𝐷) β†’ (ran 𝐹 ∩ ran 𝐺) βŠ† (𝐡 ∩ 𝐷))
122, 4, 11syl2an 597 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) β†’ (ran 𝐹 ∩ ran 𝐺) βŠ† (𝐡 ∩ 𝐷))
13 sseq0 4360 . . . . . . 7 (((ran 𝐹 ∩ ran 𝐺) βŠ† (𝐡 ∩ 𝐷) ∧ (𝐡 ∩ 𝐷) = βˆ…) β†’ (ran 𝐹 ∩ ran 𝐺) = βˆ…)
1412, 13sylan 581 . . . . . 6 (((𝐹:𝐴–1-1→𝐡 ∧ 𝐺:𝐢–1-1→𝐷) ∧ (𝐡 ∩ 𝐷) = βˆ…) β†’ (ran 𝐹 ∩ ran 𝐺) = βˆ…)
1514adantrl 715 . . . . 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 513 . . . 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 6804 . . . 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 684 . . 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 6784 . . 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 6745 . . 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 460 . 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 684 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 397   = wceq 1542   βˆͺ cun 3909   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  ran crn 5635  β€“1-1β†’wf1 6494  β€“1-1-ontoβ†’wf1o 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504
This theorem is referenced by:  undom  9004
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