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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 (φA ⊆ (𝐶 × 𝐷))
eqbrrdva.2 (φB ⊆ (𝐶 × 𝐷))
eqbrrdva.3 ((φ x 𝐶 y 𝐷) → (xAyxBy))
Assertion
Ref Expression
eqbrrdva (φA = 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
StepHypRef Expression
1 eqbrrdva.1 . . . 4 (φA ⊆ (𝐶 × 𝐷))
2 xpss 4389 . . . 4 (𝐶 × 𝐷) ⊆ (V × V)
31, 2syl6ss 2951 . . 3 (φA ⊆ (V × V))
4 df-rel 4295 . . 3 (Rel AA ⊆ (V × V))
53, 4sylibr 137 . 2 (φ → Rel A)
6 eqbrrdva.2 . . . 4 (φB ⊆ (𝐶 × 𝐷))
76, 2syl6ss 2951 . . 3 (φB ⊆ (V × V))
8 df-rel 4295 . . 3 (Rel BB ⊆ (V × V))
97, 8sylibr 137 . 2 (φ → Rel B)
101ssbrd 3796 . . . 4 (φ → (xAyx(𝐶 × 𝐷)y))
11 brxp 4318 . . . 4 (x(𝐶 × 𝐷)y ↔ (x 𝐶 y 𝐷))
1210, 11syl6ib 150 . . 3 (φ → (xAy → (x 𝐶 y 𝐷)))
136ssbrd 3796 . . . 4 (φ → (xByx(𝐶 × 𝐷)y))
1413, 11syl6ib 150 . . 3 (φ → (xBy → (x 𝐶 y 𝐷)))
15 eqbrrdva.3 . . . 4 ((φ x 𝐶 y 𝐷) → (xAyxBy))
16153expib 1106 . . 3 (φ → ((x 𝐶 y 𝐷) → (xAyxBy)))
1712, 14, 16pm5.21ndd 620 . 2 (φ → (xAyxBy))
185, 9, 17eqbrrdv 4380 1 (φA = B)
Colors of variables: wff set class
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)
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