Theorem f1we 7353

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 6774	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		frn 6713	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3		wess 5646	. . 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 6832	. . 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 7351	. . 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 1569   ⊆ wss 3904   class class class wbr 5108  {copab 5172   We wwe 5612  ran crn 5661  ⟶wf 6532  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545

This theorem is used by:  ac10ct  10025  dnwech  43803