Theorem elqsn0m 6569

Description: An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)

Assertion

Ref	Expression
elqsn0m	⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥 ∈ 𝐵)

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

Proof of Theorem elqsn0m

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2165	. 2 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅)
2		eleq2 2230	. . 3 ⊢ ([𝑦]𝑅 = 𝐵 → (𝑥 ∈ [𝑦]𝑅 ↔ 𝑥 ∈ 𝐵))
3	2	exbidv 1813	. 2 ⊢ ([𝑦]𝑅 = 𝐵 → (∃𝑥 𝑥 ∈ [𝑦]𝑅 ↔ ∃𝑥 𝑥 ∈ 𝐵))
4		eleq2 2230	. . . 4 ⊢ (dom 𝑅 = 𝐴 → (𝑦 ∈ dom 𝑅 ↔ 𝑦 ∈ 𝐴))
5	4	biimpar 295	. . 3 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ dom 𝑅)
6		ecdmn0m 6543	. . 3 ⊢ (𝑦 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝑦]𝑅)
7	5, 6	sylib 121	. 2 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 𝑥 ∈ [𝑦]𝑅)
8	1, 3, 7	ectocld 6567	1 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  dom cdm 4604  [cec 6499   / cqs 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-ec 6503  df-qs 6507

This theorem is referenced by:  elqsn0  6570  ecelqsdm  6571