Theorem dmqseqim2 38775

Description: Lemma for erimeq2 38796. (Contributed by Peter Mazsa, 29-Dec-2021.)

Assertion

Ref	Expression
dmqseqim2	⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴))))

Proof of Theorem dmqseqim2

Step	Hyp	Ref	Expression
1		dmqseqim 38774	. 2 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = ∪ 𝐴)))
2		eleq2 2822	. 2 ⊢ (ran 𝑅 = ∪ 𝐴 → (𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴))
3	1, 2	syl8 76	1 ⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴))))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ∪ cuni 4858  dom cdm 5619  ran crn 5620  Rel wrel 5624   / cqs 8627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ec 8630  df-qs 8634

This theorem is referenced by:  erimeq2  38796