Theorem eldmqsres 38507

Description: Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 21-Aug-2020.)

Assertion

Ref	Expression
eldmqsres	⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)))

Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵   𝑢,𝑅,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑢)

Proof of Theorem eldmqsres

Step	Hyp	Ref	Expression
1		elqsg 8705	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ dom (𝑅 ↾ 𝐴)𝐵 = [𝑢](𝑅 ↾ 𝐴)))
2		eldmres2 38496	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅)))
3	2	elv 3446	. . . . 5 ⊢ (𝑢 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅))
4	3	anbi1i 625	. . . 4 ⊢ ((𝑢 ∈ dom (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ ((𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)))
5		elecreseq 8688	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐴 → [𝑢](𝑅 ↾ 𝐴) = [𝑢]𝑅)
6	5	eqeq2d 2748	. . . . . . 7 ⊢ (𝑢 ∈ 𝐴 → (𝐵 = [𝑢](𝑅 ↾ 𝐴) ↔ 𝐵 = [𝑢]𝑅))
7	6	pm5.32i 574	. . . . . 6 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢]𝑅))
8	7	anbi2i 624	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴))) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢]𝑅)))
9		an21 645	. . . . 5 ⊢ (((𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴))))
10		an12 646	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢]𝑅)))
11	8, 9, 10	3bitr4i 303	. . . 4 ⊢ (((𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (𝑢 ∈ 𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)))
12	4, 11	bitri 275	. . 3 ⊢ ((𝑢 ∈ dom (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (𝑢 ∈ 𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)))
13	12	rexbii2 3080	. 2 ⊢ (∃𝑢 ∈ dom (𝑅 ↾ 𝐴)𝐵 = [𝑢](𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))
14	1, 13	bitrdi 287	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3061  Vcvv 3441  dom cdm 5625   ↾ cres 5627  [cec 8636   / cqs 8637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8640  df-qs 8644

This theorem is referenced by:  eldmqsres2  38508