Theorem 2ndresdjuf1o 33313

Description: The 2^nd function restricted to a disjoint union is a bijection. See also e.g. 2ndconst 8136. (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⟶1-1onto𝐴)

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 33312	. 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈⟶1-1𝐴)
7	1	iunfo 10647	. . 3 ⊢ (2nd ↾ 𝑈):𝑈⟶onto∪ 𝑥 ∈ 𝑋 𝐶
8		foeq3 6813	. . . 4 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 → ((2nd ↾ 𝑈):𝑈⟶onto∪ 𝑥 ∈ 𝑋 𝐶 ↔ (2nd ↾ 𝑈):𝑈⟶onto𝐴))
9	8	biimpa 482	. . 3 ⊢ ((∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴 ∧ (2nd ↾ 𝑈):𝑈⟶onto∪ 𝑥 ∈ 𝑋 𝐶) → (2nd ↾ 𝑈):𝑈⟶onto𝐴)
10	5, 7, 9	sylancl 598	. 2 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈⟶onto𝐴)
11		df-f1o 6562	. 2 ⊢ ((2nd ↾ 𝑈):𝑈⟶1-1onto𝐴 ↔ ((2nd ↾ 𝑈):𝑈⟶1-1𝐴 ∧ (2nd ↾ 𝑈):𝑈⟶onto𝐴))
12	6, 10, 11	sylanbrc 595	1 ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈⟶1-1onto𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {csn 4588  ∪ ciun 4954  Disj wdisj 5073   × cxp 5653   ↾ cres 5657  ⟶1-1wf1 6552  ⟶ontowfo 6553  ⟶1-1ontowf1o 6554  2^nd c2nd 8021

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-disj 5074  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-2nd 8023

This theorem is used by:  gsumpart  33691