Theorem eqbrrdva 4448

Description: Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)

Hypotheses

Ref	Expression
eqbrrdva.1	⊢ (φ → A ⊆ (𝐶 × 𝐷))
eqbrrdva.2	⊢ (φ → B ⊆ (𝐶 × 𝐷))
eqbrrdva.3	⊢ ((φ ∧ x ∈ 𝐶 ∧ y ∈ 𝐷) → (xAy ↔ xBy))

Assertion

Ref	Expression
eqbrrdva	⊢ (φ → A = B)

Distinct variable groups:   x,y,A   x,B,y   φ,x,y

Allowed substitution hints:   𝐶(x,y)   𝐷(x,y)

Proof of Theorem eqbrrdva

Step	Hyp	Ref	Expression
1		eqbrrdva.1	. . . 4 ⊢ (φ → A ⊆ (𝐶 × 𝐷))
2		xpss 4389	. . . 4 ⊢ (𝐶 × 𝐷) ⊆ (V × V)
3	1, 2	syl6ss 2951	. . 3 ⊢ (φ → A ⊆ (V × V))
4		df-rel 4295	. . 3 ⊢ (Rel A ↔ A ⊆ (V × V))
5	3, 4	sylibr 137	. 2 ⊢ (φ → Rel A)
6		eqbrrdva.2	. . . 4 ⊢ (φ → B ⊆ (𝐶 × 𝐷))
7	6, 2	syl6ss 2951	. . 3 ⊢ (φ → B ⊆ (V × V))
8		df-rel 4295	. . 3 ⊢ (Rel B ↔ B ⊆ (V × V))
9	7, 8	sylibr 137	. 2 ⊢ (φ → Rel B)
10	1	ssbrd 3796	. . . 4 ⊢ (φ → (xAy → x(𝐶 × 𝐷)y))
11		brxp 4318	. . . 4 ⊢ (x(𝐶 × 𝐷)y ↔ (x ∈ 𝐶 ∧ y ∈ 𝐷))
12	10, 11	syl6ib 150	. . 3 ⊢ (φ → (xAy → (x ∈ 𝐶 ∧ y ∈ 𝐷)))
13	6	ssbrd 3796	. . . 4 ⊢ (φ → (xBy → x(𝐶 × 𝐷)y))
14	13, 11	syl6ib 150	. . 3 ⊢ (φ → (xBy → (x ∈ 𝐶 ∧ y ∈ 𝐷)))
15		eqbrrdva.3	. . . 4 ⊢ ((φ ∧ x ∈ 𝐶 ∧ y ∈ 𝐷) → (xAy ↔ xBy))
16	15	3expib 1106	. . 3 ⊢ (φ → ((x ∈ 𝐶 ∧ y ∈ 𝐷) → (xAy ↔ xBy)))
17	12, 14, 16	pm5.21ndd 620	. 2 ⊢ (φ → (xAy ↔ xBy))
18	5, 9, 17	eqbrrdv 4380	1 ⊢ (φ → A = B)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911   class class class wbr 3755   × cxp 4286  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295

This theorem is referenced by: (None)