Theorem f1we 7351

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 6766	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
2		frn 6705	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
3		wess 5633	. . 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 6824	. . 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 7349	. . 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 3898   class class class wbr 5102  {copab 5166   We wwe 5599  ran crn 5648  ⟶wf 6523  –1-1→wf1 6524  –1-1-onto→wf1o 6526  ‘cfv 6527

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5542  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536

This theorem is used by:  ac10ct  10084  dnwech  43993