Theorem 2ndresdjuf1o 33006

Description: The 2^nd function restricted to a disjoint union is a bijection. See also e.g. 2ndconst 8094. (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 33005	. 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1→𝐴)
7	1	iunfo 10529	. . 3 ⊢ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶
8		foeq3 6790	. . . 4 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 → ((2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶 ↔ (2nd ↾ 𝑈):𝑈–onto→𝐴))
9	8	biimpa 481	. . 3 ⊢ ((∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→∪ 𝑥 ∈ 𝑋 𝐶) → (2nd ↾ 𝑈):𝑈–onto→𝐴)
10	5, 7, 9	sylancl 597	. 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–onto→𝐴)
11		df-f1o 6543	. 2 ⊢ ((2nd ↾ 𝑈):𝑈–1-1-onto→𝐴 ↔ ((2nd ↾ 𝑈):𝑈–1-1→𝐴 ∧ (2nd ↾ 𝑈):𝑈–onto→𝐴))
12	6, 10, 11	sylanbrc 594	1 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  {csn 4588  ∪ ciun 4955  Disj wdisj 5075   × cxp 5658   ↾ cres 5662  –1-1→wf1 6533  –onto→wfo 6534  –1-1-onto→wf1o 6535  2^nd c2nd 7983

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-disj 5076  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-2nd 7985

This theorem is used by:  gsumpart  33392