Theorem prtlem11 38859

Description: Lemma for prter2 38874. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Assertion

Ref	Expression
prtlem11	⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ ))))

Proof of Theorem prtlem11

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eceq1 8710	. . . 4 ⊢ (𝑥 = 𝐶 → [𝑥] ∼ = [𝐶] ∼ )
2	1	rspceeqv 3611	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ )
3		elqsg 8737	. . 3 ⊢ (𝐵 ∈ 𝐷 → (𝐵 ∈ (𝐴 / ∼ ) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ ))
4	2, 3	imbitrrid 246	. 2 ⊢ (𝐵 ∈ 𝐷 → ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 ∈ (𝐴 / ∼ )))
5	4	expd 415	1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ ))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  [cec 8669   / cqs 8670

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673  df-qs 8677

This theorem is referenced by:  prter2  38874