Theorem f1owe 7282

Description: Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)

Hypothesis

Ref	Expression
f1owe.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}

Assertion

Ref	Expression
f1owe	⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑆 We 𝐵 → 𝑅 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.

Step	Hyp	Ref	Expression
1		fveq2 6817	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
2	1	breq1d 5096	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑦)))
3		fveq2 6817	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
4	3	breq2d 5098	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑧)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
5		f1owe.1	. . . . 5 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}
6	2, 4, 5	brabg 5474	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
7	6	rgen2 3172	. . 3 ⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤))
8		df-isom 6485	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹:𝐴–1-1-onto→𝐵 ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤))))
9		isowe 7278	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
10	8, 9	sylbir 235	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤))) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
11	7, 10	mpan2 691	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
12	11	biimprd 248	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∀wral 3047   class class class wbr 5086  {copab 5148   We wwe 5563  –1-1-onto→wf1o 6475  ‘cfv 6476   Isom wiso 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485

This theorem is referenced by:  wemapwe  9582  dfac8b  9917  ac10ct  9920  dnwech  43081