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Theorem f1owe 7343
Description: Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)
Hypothesis
Ref Expression
f1owe.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}
Assertion
Ref Expression
f1owe (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 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.
StepHypRef Expression
1 fveq2 6882 . . . . . 6 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
21breq1d 5149 . . . . 5 (𝑥 = 𝑧 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑦)))
3 fveq2 6882 . . . . . 6 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
43breq2d 5151 . . . . 5 (𝑦 = 𝑤 → ((𝐹𝑧)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
5 f1owe.1 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}
62, 4, 5brabg 5530 . . . 4 ((𝑧𝐴𝑤𝐴) → (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
76rgen2 3189 . . 3 𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))
8 df-isom 6543 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))))
9 isowe 7339 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴𝑆 We 𝐵))
108, 9sylbir 234 . . 3 ((𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))) → (𝑅 We 𝐴𝑆 We 𝐵))
117, 10mpan2 688 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝑅 We 𝐴𝑆 We 𝐵))
1211biimprd 247 1 (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wral 3053   class class class wbr 5139  {copab 5201   We wwe 5621  1-1-ontowf1o 6533  cfv 6534   Isom wiso 6535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543
This theorem is referenced by:  wemapwe  9689  dfac8b  10023  ac10ct  10026  dnwech  42340
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