Theorem f1owe 7351

Description: Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)

Hypothesis

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

Assertion

Ref	Expression
f1owe	⊢ (𝐹:𝐴–1-1-onto→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 f1owe

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
2	1	breq1d 5134	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑦)))
3		fveq2 6881	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
4	3	breq2d 5136	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑧)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
5		f1owe.1	. . . . 5 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}
6	2, 4, 5	brabg 5519	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
7	6	rgen2 3185	. . 3 ⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤))
8		df-isom 6545	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹:𝐴–1-1-onto→𝐵 ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤))))
9		isowe 7347	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
10	8, 9	sylbir 235	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤))) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
11	7, 10	mpan2 691	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
12	11	biimprd 248	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∀wral 3052   class class class wbr 5124  {copab 5186   We wwe 5610  –1-1-onto→wf1o 6535  ‘cfv 6536   Isom wiso 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  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 6489  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 referenced by:  wemapwe  9716  dfac8b  10050  ac10ct  10053  dnwech  43047