Theorem prtlem11 34151

Description: Lemma for prter2 34166. (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 3062	. . . 4 ⊢ (𝐶 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐶)
2		r19.41v 3089	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ) ↔ (∃𝑥 ∈ 𝐴 𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ))
3		eceq1 7782	. . . . . . 7 ⊢ (𝑥 = 𝐶 → [𝑥] ∼ = [𝐶] ∼ )
4		eqtr3 2643	. . . . . . . 8 ⊢ (([𝑥] ∼ = [𝐶] ∼ ∧ 𝐵 = [𝐶] ∼ ) → [𝑥] ∼ = 𝐵)
5	4	eqcomd 2628	. . . . . . 7 ⊢ (([𝑥] ∼ = [𝐶] ∼ ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 = [𝑥] ∼ )
6	3, 5	sylan 488	. . . . . 6 ⊢ ((𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 = [𝑥] ∼ )
7	6	reximi 3011	. . . . 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 489	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ )
10		elqsg 7798	. . 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 452	1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ ))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  [cec 7740   / cqs 7741

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-3an 1039  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  prter2  34166