Theorem prtlem9 39436

Description: Lemma for prter3 39454. (Contributed by Rodolfo Medina, 25-Sep-2010.)

Assertion

Ref	Expression
prtlem9	⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   ∼ (𝑥)

Proof of Theorem prtlem9

Step	Hyp	Ref	Expression
1		risset 3231	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
2		eceq1 8706	. . 3 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ )
3	2	reximi 3094	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝑥 = 𝐴 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )
4	1, 3	sylbi 219	1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ 𝐵 [𝑥] ∼ = [𝐴] ∼ )

Syntax hints:   → wi 4   = wceq 1554   ∈ wcel 2136  ∃wrex 3080  [cec 8664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-xp 5646  df-cnv 5648  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-ec 8668

This theorem is referenced by: (None)