Theorem wessf1orn 45372

Description: Given a function 𝐹 on a well-ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
wessf1orn.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
wessf1orn.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
wessf1orn.r	⊢ (𝜑 → 𝑅 We 𝐴)

Assertion

Ref	Expression
wessf1orn	⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑅

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem wessf1orn

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wessf1orn.f	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		wessf1orn.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		wessf1orn.r	. 2 ⊢ (𝜑 → 𝑅 We 𝐴)
4		eqid 2734	. 2 ⊢ (𝑦 ∈ ran 𝐹 ↦ (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥)) = (𝑦 ∈ ran 𝐹 ↦ (℩𝑥 ∈ (◡𝐹 “ {𝑦})∀𝑧 ∈ (◡𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥))
5	1, 2, 3, 4	wessf1ornlem 45371	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  𝒫 cpw 4552  {csn 4578   class class class wbr 5096   ↦ cmpt 5177   We wwe 5574  ^◡ccnv 5621  ran crn 5623   ↾ cres 5624   “ cima 5625   Fn wfn 6485  –1-1-onto→wf1o 6489  ℩crio 7312

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313

This theorem is referenced by:  ssnnf1octb  45380  sge0resrn  46590  nnfoctbdj  46642