Theorem wessf1orn 44370

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 2724	. 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 44369	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2098  ∀wral 3053  ∃wrex 3062  𝒫 cpw 4594  {csn 4620   class class class wbr 5138   ↦ cmpt 5221   We wwe 5620  ^◡ccnv 5665  ran crn 5667   ↾ cres 5668   “ cima 5669   Fn wfn 6528  –1-1-onto→wf1o 6532  ℩crio 7356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357

This theorem is referenced by:  ssnnf1octb  44378  sge0resrn  45605  nnfoctbdj  45657