Theorem isoresbr 5777

Description: A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)

Assertion

Ref	Expression
isoresbr	⊢ ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Proof of Theorem isoresbr

Step	Hyp	Ref	Expression
1		isorel 5776	. . . 4 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦)))
2		fvres 5510	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
3		fvres 5510	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
4	2, 3	breqan12d 3998	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
5	4	adantl 275	. . . 4 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
6	1, 5	bitrd 187	. . 3 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
7	6	biimpd 143	. 2 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
8	7	ralrimivva 2548	1 ⊢ ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2136  ∀wral 2444   class class class wbr 3982   ↾ cres 4606   “ cima 4607  ‘cfv 5188   Isom wiso 5189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-res 4616  df-iota 5153  df-fv 5196  df-isom 5197

This theorem is referenced by: (None)