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Theorem f1owe 6588
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 6178 . . . . . 6 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
21breq1d 4654 . . . . 5 (𝑥 = 𝑧 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑦)))
3 fveq2 6178 . . . . . 6 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
43breq2d 4656 . . . . 5 (𝑦 = 𝑤 → ((𝐹𝑧)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
5 f1owe.1 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}
62, 4, 5brabg 4984 . . . 4 ((𝑧𝐴𝑤𝐴) → (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
76rgen2a 2974 . . 3 𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))
8 df-isom 5885 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))))
9 isowe 6584 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴𝑆 We 𝐵))
108, 9sylbir 225 . . 3 ((𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))) → (𝑅 We 𝐴𝑆 We 𝐵))
117, 10mpan2 706 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝑅 We 𝐴𝑆 We 𝐵))
1211biimprd 238 1 (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wral 2909   class class class wbr 4644  {copab 4703   We wwe 5062  1-1-ontowf1o 5875  cfv 5876   Isom wiso 5877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885
This theorem is referenced by:  wemapwe  8579  dfac8b  8839  ac10ct  8842  dnwech  37437
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