Theorem 2ndresdjuf1o 32624

Description: The 2^nd function restricted to a disjoint union is a bijection. See also e.g. 2ndconst 8026. (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 32623	. 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1→𝐴)
7	1	iunfo 10425	. . 3 ⊢ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶
8		foeq3 6728	. . . 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 586	. 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–onto→𝐴)
11		df-f1o 6483	. 2 ⊢ ((2nd ↾ 𝑈):𝑈–1-1-onto→𝐴 ↔ ((2nd ↾ 𝑈):𝑈–1-1→𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→𝐴))
12	6, 10, 11	sylanbrc 583	1 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  {csn 4571  ∪ ciun 4936  Disj wdisj 5053   × cxp 5609   ↾ cres 5613  –1-1→wf1 6473  –onto→wfo 6474  –1-1-onto→wf1o 6475  2^nd c2nd 7915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-disj 5054  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-2nd 7917

This theorem is referenced by:  gsumpart  33029