Theorem sspred 6330

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 4223	. 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
2		df-pred 6321	. . . 4 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3	2	sseq1i 4012	. . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐵)
4		dfss2 3969	. . 3 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐵 ↔ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
5		in32 4230	. . . 4 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋}))
6	5	eqeq1i 2742	. . 3 ⊢ (((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (◡𝑅 “ {𝑋})) ↔ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
7	3, 4, 6	3bitri 297	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
8		ineq1 4213	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 → ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐵 ∩ (◡𝑅 “ {𝑋})))
9	8	eqeq1d 2739	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 → (((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})) ↔ (𝐵 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))))
10	9	biimpa 476	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = 𝐵 ∧ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))) → (𝐵 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
11		df-pred 6321	. . . 4 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
12	10, 11, 2	3eqtr4g 2802	. . 3 ⊢ (((𝐴 ∩ 𝐵) = 𝐵 ∧ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))) → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))
13	12	eqcomd 2743	. 2 ⊢ (((𝐴 ∩ 𝐵) = 𝐵 ∧ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
14	1, 7, 13	syl2anb 598	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∩ cin 3950   ⊆ wss 3951  {csn 4626  ^◡ccnv 5684   “ cima 5688  Predcpred 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pred 6321

This theorem is referenced by:  frmin  9789