Theorem prtlem11 34470

Description: Lemma for prter2 34485. (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		risset 3091	. . . 4 ⊢ (𝐶 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐶)
2		r19.41v 3118	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ) ↔ (∃𝑥 ∈ 𝐴 𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ))
3		eceq1 7825	. . . . . . 7 ⊢ (𝑥 = 𝐶 → [𝑥] ∼ = [𝐶] ∼ )
4		eqtr3 2672	. . . . . . . 8 ⊢ (([𝑥] ∼ = [𝐶] ∼ ∧ 𝐵 = [𝐶] ∼ ) → [𝑥] ∼ = 𝐵)
5	4	eqcomd 2657	. . . . . . 7 ⊢ (([𝑥] ∼ = [𝐶] ∼ ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 = [𝑥] ∼ )
6	3, 5	sylan 487	. . . . . 6 ⊢ ((𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 = [𝑥] ∼ )
7	6	reximi 3040	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ )
8	2, 7	sylbir 225	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ )
9	1, 8	sylanb 488	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ )
10		elqsg 7841	. . 3 ⊢ (𝐵 ∈ 𝐷 → (𝐵 ∈ (𝐴 / ∼ ) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ ))
11	9, 10	syl5ibr 236	. 2 ⊢ (𝐵 ∈ 𝐷 → ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 ∈ (𝐴 / ∼ )))
12	11	expd 451	1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ ))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∃wrex 2942  [cec 7785   / cqs 7786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ec 7789  df-qs 7793

This theorem is referenced by:  prter2  34485