Theorem eldmqsres2 36045

Description: Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 22-Aug-2020.)

Assertion

Ref	Expression
eldmqsres2	⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅))

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

Allowed substitution hints:   𝑉(𝑥,𝑢)

Proof of Theorem eldmqsres2

Step	Hyp	Ref	Expression
1		eldmqsres 36044	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)))
2		df-rex 3059	. . . 4 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))
3		19.41v 1957	. . . 4 ⊢ (∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))
4	2, 3	bitri 278	. . 3 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))
5	4	rexbii 3161	. 2 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))
6	1, 5	bitr4di 292	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542  ∃wex 1786   ∈ wcel 2114  ∃wrex 3054  dom cdm 5525   ↾ cres 5527  [cec 8318   / cqs 8319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ec 8322  df-qs 8326

This theorem is referenced by:  releldmqs  36393