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Theorem sspred 5726
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 3850 . 2 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
2 df-pred 5718 . . . 4 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
32sseq1i 3662 . . 3 (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐵)
4 df-ss 3621 . . 3 ((𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐵 ↔ ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (𝑅 “ {𝑋})))
5 in32 3858 . . . 4 ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = ((𝐴𝐵) ∩ (𝑅 “ {𝑋}))
65eqeq1i 2656 . . 3 (((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (𝑅 “ {𝑋})) ↔ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
73, 4, 63bitri 286 . 2 (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
8 ineq1 3840 . . . . . 6 ((𝐴𝐵) = 𝐵 → ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐵 ∩ (𝑅 “ {𝑋})))
98eqeq1d 2653 . . . . 5 ((𝐴𝐵) = 𝐵 → (((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})) ↔ (𝐵 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))))
109biimpa 500 . . . 4 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → (𝐵 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
11 df-pred 5718 . . . 4 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
1210, 11, 23eqtr4g 2710 . . 3 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))
1312eqcomd 2657 . 2 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
141, 7, 13syl2anb 495 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 383   = wceq 1523  cin 3606  wss 3607  {csn 4210  ccnv 5142  cima 5146  Predcpred 5717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-pred 5718
This theorem is referenced by:  frmin  31867
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