Theorem wessf1orn 45093

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 2740	. 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 45092	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  𝒫 cpw 4622  {csn 4648   class class class wbr 5166   ↦ cmpt 5249   We wwe 5651  ^◡ccnv 5699  ran crn 5701   ↾ cres 5702   “ cima 5703   Fn wfn 6568  –1-1-onto→wf1o 6572  ℩crio 7403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404

This theorem is referenced by:  ssnnf1octb  45101  sge0resrn  46325  nnfoctbdj  46377