Theorem brres2 34035

Description: Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.) (Revised by Peter Mazsa, 16-Dec-2021.)

Assertion

Ref	Expression
brres2	⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ 𝐵(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝐶)

Proof of Theorem brres2

Step	Hyp	Ref	Expression
1		brresALTV 34032	. . 3 ⊢ (𝐶 ∈ ran (𝑅 ↾ 𝐴) → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))
2	1	pm5.32i 669	. 2 ⊢ ((𝐶 ∈ ran (𝑅 ↾ 𝐴) ∧ 𝐵(𝑅 ↾ 𝐴)𝐶) ↔ (𝐶 ∈ ran (𝑅 ↾ 𝐴) ∧ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))
3		relres 5426	. . . 4 ⊢ Rel (𝑅 ↾ 𝐴)
4	3	relelrni 5363	. . 3 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 → 𝐶 ∈ ran (𝑅 ↾ 𝐴))
5	4	pm4.71ri 665	. 2 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐶 ∈ ran (𝑅 ↾ 𝐴) ∧ 𝐵(𝑅 ↾ 𝐴)𝐶))
6		brinxp2ALTV 34034	. . 3 ⊢ (𝐵(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝐶 ↔ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ran (𝑅 ↾ 𝐴)) ∧ 𝐵𝑅𝐶))
7		df-3an 1039	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ran (𝑅 ↾ 𝐴) ∧ 𝐵𝑅𝐶) ↔ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ran (𝑅 ↾ 𝐴)) ∧ 𝐵𝑅𝐶))
8		3anan12 1051	. . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ran (𝑅 ↾ 𝐴) ∧ 𝐵𝑅𝐶) ↔ (𝐶 ∈ ran (𝑅 ↾ 𝐴) ∧ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))
9	6, 7, 8	3bitr2i 288	. 2 ⊢ (𝐵(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝐶 ↔ (𝐶 ∈ ran (𝑅 ↾ 𝐴) ∧ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶)))
10	2, 5, 9	3bitr4i 292	1 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ 𝐵(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝐶)

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990   ∩ cin 3573   class class class wbr 4653   × cxp 5112  ran crn 5115   ↾ cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  brinxprnres  34059