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Theorem sspred 5591
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 3778 . 2 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
2 df-pred 5583 . . . 4 Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
32sseq1i 3591 . . 3 (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ (𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐵)
4 df-ss 3553 . . 3 ((𝐴 ∩ (𝑅 “ {𝑋})) ⊆ 𝐵 ↔ ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (𝑅 “ {𝑋})))
5 in32 3786 . . . 4 ((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = ((𝐴𝐵) ∩ (𝑅 “ {𝑋}))
65eqeq1i 2614 . . 3 (((𝐴 ∩ (𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (𝑅 “ {𝑋})) ↔ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
73, 4, 63bitri 284 . 2 (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
8 ineq1 3768 . . . . . 6 ((𝐴𝐵) = 𝐵 → ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐵 ∩ (𝑅 “ {𝑋})))
98eqeq1d 2611 . . . . 5 ((𝐴𝐵) = 𝐵 → (((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})) ↔ (𝐵 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))))
109biimpa 499 . . . 4 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → (𝐵 ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋})))
11 df-pred 5583 . . . 4 Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (𝑅 “ {𝑋}))
1210, 11, 23eqtr4g 2668 . . 3 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))
1312eqcomd 2615 . 2 (((𝐴𝐵) = 𝐵 ∧ ((𝐴𝐵) ∩ (𝑅 “ {𝑋})) = (𝐴 ∩ (𝑅 “ {𝑋}))) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
141, 7, 13syl2anb 494 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 382   = wceq 1474  cin 3538  wss 3539  {csn 4124  ccnv 5027  cima 5031  Predcpred 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-in 3546  df-ss 3553  df-pred 5583
This theorem is referenced by:  frmin  30817
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