Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2ndresdjuf1o Structured version   Visualization version   GIF version

Theorem 2ndresdjuf1o 31870
Description: The 2nd function restricted to a disjoint union is a bijection. See also e.g. 2ndconst 8086. (Contributed by Thierry Arnoux, 23-Jun-2024.)
Hypotheses
Ref Expression
2ndresdju.u 𝑈 = 𝑥𝑋 ({𝑥} × 𝐶)
2ndresdju.a (𝜑𝐴𝑉)
2ndresdju.x (𝜑𝑋𝑊)
2ndresdju.1 (𝜑Disj 𝑥𝑋 𝐶)
2ndresdju.2 (𝜑 𝑥𝑋 𝐶 = 𝐴)
Assertion
Ref Expression
2ndresdjuf1o (𝜑 → (2nd𝑈):𝑈1-1-onto𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝜑,𝑥
Allowed substitution hints:   𝐶(𝑥)   𝑈(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem 2ndresdjuf1o
StepHypRef Expression
1 2ndresdju.u . . 3 𝑈 = 𝑥𝑋 ({𝑥} × 𝐶)
2 2ndresdju.a . . 3 (𝜑𝐴𝑉)
3 2ndresdju.x . . 3 (𝜑𝑋𝑊)
4 2ndresdju.1 . . 3 (𝜑Disj 𝑥𝑋 𝐶)
5 2ndresdju.2 . . 3 (𝜑 𝑥𝑋 𝐶 = 𝐴)
61, 2, 3, 4, 52ndresdju 31869 . 2 (𝜑 → (2nd𝑈):𝑈1-1𝐴)
71iunfo 10533 . . 3 (2nd𝑈):𝑈onto 𝑥𝑋 𝐶
8 foeq3 6803 . . . 4 ( 𝑥𝑋 𝐶 = 𝐴 → ((2nd𝑈):𝑈onto 𝑥𝑋 𝐶 ↔ (2nd𝑈):𝑈onto𝐴))
98biimpa 477 . . 3 (( 𝑥𝑋 𝐶 = 𝐴 ∧ (2nd𝑈):𝑈onto 𝑥𝑋 𝐶) → (2nd𝑈):𝑈onto𝐴)
105, 7, 9sylancl 586 . 2 (𝜑 → (2nd𝑈):𝑈onto𝐴)
11 df-f1o 6550 . 2 ((2nd𝑈):𝑈1-1-onto𝐴 ↔ ((2nd𝑈):𝑈1-1𝐴 ∧ (2nd𝑈):𝑈onto𝐴))
126, 10, 11sylanbrc 583 1 (𝜑 → (2nd𝑈):𝑈1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {csn 4628   ciun 4997  Disj wdisj 5113   × cxp 5674  cres 5678  1-1wf1 6540  ontowfo 6541  1-1-ontowf1o 6542  2nd c2nd 7973
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-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
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-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-2nd 7975
This theorem is referenced by:  gsumpart  32202
  Copyright terms: Public domain W3C validator