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Theorem f1owe 6503
 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 6103 . . . . . 6 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
21breq1d 4593 . . . . 5 (𝑥 = 𝑧 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑦)))
3 fveq2 6103 . . . . . 6 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
43breq2d 4595 . . . . 5 (𝑦 = 𝑤 → ((𝐹𝑧)𝑆(𝐹𝑦) ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
5 f1owe.1 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}
62, 4, 5brabg 4919 . . . 4 ((𝑧𝐴𝑤𝐴) → (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤)))
76rgen2a 2960 . . 3 𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))
8 df-isom 5813 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))))
9 isowe 6499 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴𝑆 We 𝐵))
108, 9sylbir 224 . . 3 ((𝐹:𝐴1-1-onto𝐵 ∧ ∀𝑧𝐴𝑤𝐴 (𝑧𝑅𝑤 ↔ (𝐹𝑧)𝑆(𝐹𝑤))) → (𝑅 We 𝐴𝑆 We 𝐵))
117, 10mpan2 703 . 2 (𝐹:𝐴1-1-onto𝐵 → (𝑅 We 𝐴𝑆 We 𝐵))
1211biimprd 237 1 (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∀wral 2896   class class class wbr 4583  {copab 4642   We wwe 4996  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813 This theorem is referenced by:  wemapwe  8477  dfac8b  8737  ac10ct  8740  dnwech  36636
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