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Theorem 2ndresdjuf1o 30415
 Description: The 2nd function restricted to a disjoint union is a bijection. See also e.g. 2ndconst 7783. (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 30414 . 2 (𝜑 → (2nd𝑈):𝑈1-1𝐴)
71iunfo 9954 . . 3 (2nd𝑈):𝑈onto 𝑥𝑋 𝐶
8 foeq3 6567 . . . 4 ( 𝑥𝑋 𝐶 = 𝐴 → ((2nd𝑈):𝑈onto 𝑥𝑋 𝐶 ↔ (2nd𝑈):𝑈onto𝐴))
98biimpa 480 . . 3 (( 𝑥𝑋 𝐶 = 𝐴 ∧ (2nd𝑈):𝑈onto 𝑥𝑋 𝐶) → (2nd𝑈):𝑈onto𝐴)
105, 7, 9sylancl 589 . 2 (𝜑 → (2nd𝑈):𝑈onto𝐴)
11 df-f1o 6335 . 2 ((2nd𝑈):𝑈1-1-onto𝐴 ↔ ((2nd𝑈):𝑈1-1𝐴 ∧ (2nd𝑈):𝑈onto𝐴))
126, 10, 11sylanbrc 586 1 (𝜑 → (2nd𝑈):𝑈1-1-onto𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  {csn 4528  ∪ ciun 4884  Disj wdisj 4998   × cxp 5521   ↾ cres 5525  –1-1→wf1 6325  –onto→wfo 6326  –1-1-onto→wf1o 6327  2nd c2nd 7674 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-disj 4999  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-2nd 7676 This theorem is referenced by:  gsumpart  30743
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