Theorem sspred 6150

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 4191	. 2 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
2		df-pred 6142	. . . 4 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
3	2	sseq1i 3994	. . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐵)
4		df-ss 3951	. . 3 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐵 ↔ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
5		in32 4197	. . . 4 ⊢ ((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋}))
6	5	eqeq1i 2826	. . 3 ⊢ (((𝐴 ∩ (◡𝑅 “ {𝑋})) ∩ 𝐵) = (𝐴 ∩ (◡𝑅 “ {𝑋})) ↔ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
7	3, 4, 6	3bitri 299	. 2 ⊢ (Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
8		ineq1 4180	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 → ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐵 ∩ (◡𝑅 “ {𝑋})))
9	8	eqeq1d 2823	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 → (((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})) ↔ (𝐵 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))))
10	9	biimpa 479	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = 𝐵 ∧ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))) → (𝐵 ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋})))
11		df-pred 6142	. . . 4 ⊢ Pred(𝑅, 𝐵, 𝑋) = (𝐵 ∩ (◡𝑅 “ {𝑋}))
12	10, 11, 2	3eqtr4g 2881	. . 3 ⊢ (((𝐴 ∩ 𝐵) = 𝐵 ∧ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))) → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋))
13	12	eqcomd 2827	. 2 ⊢ (((𝐴 ∩ 𝐵) = 𝐵 ∧ ((𝐴 ∩ 𝐵) ∩ (◡𝑅 “ {𝑋})) = (𝐴 ∩ (◡𝑅 “ {𝑋}))) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
14	1, 7, 13	syl2anb 599	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∩ cin 3934   ⊆ wss 3935  {csn 4560  ^◡ccnv 5548   “ cima 5552  Predcpred 6141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-in 3942  df-ss 3951  df-pred 6142

This theorem is referenced by:  frmin  33079