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Theorem sspred 5875
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 3981 . 2 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
2 df-pred 5867 . . . 4 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
32sseq1i 3791 . . 3 (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐵)
4 df-ss 3748 . . 3 ((𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐵 ↔ ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (𝑅 “ {𝑋})))
5 in32 3987 . . . 4 ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = ((𝐴𝐵) ∩ (𝑅 “ {𝑋}))
65eqeq1i 2770 . . 3 (((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (𝑅 “ {𝑋})) ↔ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
73, 4, 63bitri 288 . 2 (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
8 ineq1 3971 . . . . . 6 ((𝐴𝐵) = 𝐵 → ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐵 ∩ (𝑅 “ {𝑋})))
98eqeq1d 2767 . . . . 5 ((𝐴𝐵) = 𝐵 → (((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})) ↔ (𝐵 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))))
109biimpa 468 . . . 4 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → (𝐵 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
11 df-pred 5867 . . . 4 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
1210, 11, 23eqtr4g 2824 . . 3 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))
1312eqcomd 2771 . 2 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
141, 7, 13syl2anb 591 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 384   = wceq 1652  cin 3733  wss 3734  {csn 4336  ccnv 5278  cima 5282  Predcpred 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3741  df-ss 3748  df-pred 5867
This theorem is referenced by:  frmin  32207
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