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Theorem f1owe 7089
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 6649 . . . . . 6 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
21breq1d 5043 . . . . 5 (𝑥 = 𝑧 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑦)))
3 fveq2 6649 . . . . . 6 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
43breq2d 5045 . . . . 5 (𝑦 = 𝑤 → ((𝐹𝑧)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
5 f1owe.1 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}
62, 4, 5brabg 5394 . . . 4 ((𝑧𝐴𝑤𝐴) → (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
76rgen2 3171 . . 3 𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))
8 df-isom 6337 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))))
9 isowe 7085 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴𝑆 We 𝐵))
108, 9sylbir 238 . . 3 ((𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))) → (𝑅 We 𝐴𝑆 We 𝐵))
117, 10mpan2 690 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝑅 We 𝐴𝑆 We 𝐵))
1211biimprd 251 1 (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wral 3109   class class class wbr 5033  {copab 5095   We wwe 5481  1-1-ontowf1o 6327  cfv 6328   Isom wiso 6329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337
This theorem is referenced by:  wemapwe  9148  dfac8b  9446  ac10ct  9449  dnwech  39989
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