Theorem sspred 5688

Description: Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)

Assertion

Ref	Expression
sspred	⊢ ((𝐵 ⊆ 𝐴 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem sspred

Step	Hyp	Ref	Expression
1		sseqin2 3817	. 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
2		df-pred 5680	. . . 4 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3	2	sseq1i 3629	. . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐵)
4		df-ss 3588	. . 3 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐵 ↔ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
5		in32 3825	. . . 4 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋}))
6	5	eqeq1i 2627	. . 3 ⊢ (((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (◡𝑅 “ {𝑋})) ↔ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
7	3, 4, 6	3bitri 286	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
8		ineq1 3807	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 → ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐵 ∩ (◡𝑅 “ {𝑋})))
9	8	eqeq1d 2624	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 → (((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})) ↔ (𝐵 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))))
10	9	biimpa 501	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = 𝐵 ∧ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))) → (𝐵 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
11		df-pred 5680	. . . 4 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
12	10, 11, 2	3eqtr4g 2681	. . 3 ⊢ (((𝐴 ∩ 𝐵) = 𝐵 ∧ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))) → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))
13	12	eqcomd 2628	. 2 ⊢ (((𝐴 ∩ 𝐵) = 𝐵 ∧ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
14	1, 7, 13	syl2anb 496	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∩ cin 3573   ⊆ wss 3574  {csn 4177  ^◡ccnv 5113   “ cima 5117  Predcpred 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pred 5680

This theorem is referenced by:  frmin  31739