Theorem 2ndresdjuf1o 30888

Description: The 2^nd function restricted to a disjoint union is a bijection. See also e.g. 2ndconst 7912. (Contributed by Thierry Arnoux, 23-Jun-2024.)

Hypotheses

Ref	Expression
2ndresdju.u	⊢ 𝑈 = ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)
2ndresdju.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
2ndresdju.x	⊢ (𝜑 → 𝑋 ∈ 𝑊)
2ndresdju.1	⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝐶)
2ndresdju.2	⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴)

Assertion

Ref	Expression
2ndresdjuf1o	⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝜑,𝑥

Allowed substitution hints:   𝐶(𝑥)   𝑈(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem 2ndresdjuf1o

Step	Hyp	Ref	Expression
1		2ndresdju.u	. . 3 ⊢ 𝑈 = ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)
2		2ndresdju.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		2ndresdju.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
4		2ndresdju.1	. . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝐶)
5		2ndresdju.2	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴)
6	1, 2, 3, 4, 5	2ndresdju 30887	. 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1→𝐴)
7	1	iunfo 10226	. . 3 ⊢ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶
8		foeq3 6670	. . . 4 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 → ((2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶 ↔ (2nd ↾ 𝑈):𝑈–onto→𝐴))
9	8	biimpa 476	. . 3 ⊢ ((∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶) → (2nd ↾ 𝑈):𝑈–onto→𝐴)
10	5, 7, 9	sylancl 585	. 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–onto→𝐴)
11		df-f1o 6425	. 2 ⊢ ((2nd ↾ 𝑈):𝑈–1-1-onto→𝐴 ↔ ((2nd ↾ 𝑈):𝑈–1-1→𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→𝐴))
12	6, 10, 11	sylanbrc 582	1 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {csn 4558  ∪ ciun 4921  Disj wdisj 5035   × cxp 5578   ↾ cres 5582  –1-1→wf1 6415  –onto→wfo 6416  –1-1-onto→wf1o 6417  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-2nd 7805

This theorem is referenced by:  gsumpart  31217