Theorem f1owe 7108

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 6672	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
2	1	breq1d 5078	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑦)))
3		fveq2 6672	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
4	3	breq2d 5080	. . . . 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 5428	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
7	6	rgen2 3205	. . 3 ⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤))
8		df-isom 6366	. . . 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 7104	. . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
10	8, 9	sylbir 237	. . 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 689	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
12	11	biimprd 250	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∀wral 3140   class class class wbr 5068  {copab 5130   We wwe 5515  –1-1-onto→wf1o 6356  ‘cfv 6357   Isom wiso 6358

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366

This theorem is referenced by:  wemapwe  9162  dfac8b  9459  ac10ct  9462  dnwech  39655