Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  f1ocof1ob2 Structured version   Visualization version   GIF version

Theorem f1ocof1ob2 47094
Description: If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. Symmetric version of f1ocof1ob 47093 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 7-Oct-2024.)
Assertion
Ref Expression
f1ocof1ob2 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1-onto𝐷 ↔ (𝐹:𝐴1-1-onto𝐶𝐺:𝐶1-1-onto𝐷)))

Proof of Theorem f1ocof1ob2
StepHypRef Expression
1 f1ocof1ob 47093 . 2 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1-onto𝐷 ↔ (𝐹:𝐴1-1𝐶𝐺:𝐶1-1-onto𝐷)))
2 f1f1orn 6859 . . . . . 6 (𝐹:𝐴1-1𝐶𝐹:𝐴1-1-onto→ran 𝐹)
3 f1oeq3 6838 . . . . . 6 (ran 𝐹 = 𝐶 → (𝐹:𝐴1-1-onto→ran 𝐹𝐹:𝐴1-1-onto𝐶))
42, 3imbitrid 244 . . . . 5 (ran 𝐹 = 𝐶 → (𝐹:𝐴1-1𝐶𝐹:𝐴1-1-onto𝐶))
543ad2ant3 1136 . . . 4 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴1-1𝐶𝐹:𝐴1-1-onto𝐶))
6 f1of1 6847 . . . 4 (𝐹:𝐴1-1-onto𝐶𝐹:𝐴1-1𝐶)
75, 6impbid1 225 . . 3 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴1-1𝐶𝐹:𝐴1-1-onto𝐶))
87anbi1d 631 . 2 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐹:𝐴1-1𝐶𝐺:𝐶1-1-onto𝐷) ↔ (𝐹:𝐴1-1-onto𝐶𝐺:𝐶1-1-onto𝐷)))
91, 8bitrd 279 1 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1-onto𝐷 ↔ (𝐹:𝐴1-1-onto𝐶𝐺:𝐶1-1-onto𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  ran crn 5686  ccom 5689  wf 6557  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-11 2157  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-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  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-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator