Theorem f1we 7359

Description: Pull back a well ordering by a one-to-one function. (Contributed by Eric Schmidt, 4-Aug-2026.)

Hypothesis

Ref	Expression
f1owe.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}

Assertion

Ref	Expression
f1we	⊢ (𝐹:𝐴–1-1→𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))

Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦)   𝐵(𝑥, 𝑦)   𝑅(𝑥, 𝑦)

Proof of Theorem f1we

Step	Hyp	Ref	Expression
1		f1f 6775	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		frn 6714	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3		wess 5645	. . 3 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝑆 We 𝐵 → 𝑆 We ran 𝐹))
4	1, 2, 3	3syl 19	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝑆 We 𝐵 → 𝑆 We ran 𝐹))
5		f1f1orn 6833	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
6		f1owe.1	. . . 4 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}
7	6	f1owe 7357	. . 3 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 → (𝑅 We 𝐴 ↔ 𝑆 We ran 𝐹))
8	5, 7	syl 18	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝑅 We 𝐴 ↔ 𝑆 We ran 𝐹))
9	4, 8	sylibrd 262	1 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ⊆ wss 3902   class class class wbr 5107  {copab 5171   We wwe 5611  ran crn 5660  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546

This theorem is used by:  ac10ct  10040  dnwech  43876