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Theorem f1owe 6600
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 6189 . . . . . 6 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
21breq1d 4661 . . . . 5 (𝑥 = 𝑧 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑦)))
3 fveq2 6189 . . . . . 6 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
43breq2d 4663 . . . . 5 (𝑦 = 𝑤 → ((𝐹𝑧)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
5 f1owe.1 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}
62, 4, 5brabg 4992 . . . 4 ((𝑧𝐴𝑤𝐴) → (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
76rgen2a 2976 . . 3 𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))
8 df-isom 5895 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))))
9 isowe 6596 . . . 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 707 . 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 1482  wral 2911   class class class wbr 4651  {copab 4710   We wwe 5070  1-1-ontowf1o 5885  cfv 5886   Isom wiso 5887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895
This theorem is referenced by:  wemapwe  8591  dfac8b  8851  ac10ct  8854  dnwech  37444
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