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Theorem sspred 6149
Description: Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)
Assertion
Ref Expression
sspred ((𝐵𝐴 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem sspred
StepHypRef Expression
1 sseqin2 4189 . 2 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
2 df-pred 6141 . . . 4 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
32sseq1i 3992 . . 3 (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐵)
4 df-ss 3949 . . 3 ((𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐵 ↔ ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (𝑅 “ {𝑋})))
5 in32 4195 . . . 4 ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = ((𝐴𝐵) ∩ (𝑅 “ {𝑋}))
65eqeq1i 2823 . . 3 (((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (𝑅 “ {𝑋})) ↔ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
73, 4, 63bitri 298 . 2 (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
8 ineq1 4178 . . . . . 6 ((𝐴𝐵) = 𝐵 → ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐵 ∩ (𝑅 “ {𝑋})))
98eqeq1d 2820 . . . . 5 ((𝐴𝐵) = 𝐵 → (((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})) ↔ (𝐵 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))))
109biimpa 477 . . . 4 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → (𝐵 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
11 df-pred 6141 . . . 4 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
1210, 11, 23eqtr4g 2878 . . 3 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))
1312eqcomd 2824 . 2 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
141, 7, 13syl2anb 597 1 ((𝐵𝐴 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  cin 3932  wss 3933  {csn 4557  ccnv 5547  cima 5551  Predcpred 6140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949  df-pred 6141
This theorem is referenced by:  frmin  32981
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