Theorem f1owe 7374

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 6905	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
2	1	breq1d 5152	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑦)))
3		fveq2 6905	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
4	3	breq2d 5154	. . . . 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 5543	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
7	6	rgen2 3198	. . 3 ⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤))
8		df-isom 6569	. . . 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 7370	. . . 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 1539  ∀wral 3060   class class class wbr 5142  {copab 5204   We wwe 5635  –1-1-onto→wf1o 6559  ‘cfv 6560   Isom wiso 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569

This theorem is referenced by:  wemapwe  9738  dfac8b  10072  ac10ct  10075  dnwech  43065