Theorem elqsn0m 6427

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 2100	. 2 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅)
2		eleq2 2163	. . 3 ⊢ ([𝑦]𝑅 = 𝐵 → (𝑥 ∈ [𝑦]𝑅 ↔ 𝑥 ∈ 𝐵))
3	2	exbidv 1764	. 2 ⊢ ([𝑦]𝑅 = 𝐵 → (∃𝑥 𝑥 ∈ [𝑦]𝑅 ↔ ∃𝑥 𝑥 ∈ 𝐵))
4		eleq2 2163	. . . 4 ⊢ (dom 𝑅 = 𝐴 → (𝑦 ∈ dom 𝑅 ↔ 𝑦 ∈ 𝐴))
5	4	biimpar 293	. . 3 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ dom 𝑅)
6		ecdmn0m 6401	. . 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 6425	1 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1299  ∃wex 1436   ∈ wcel 1448  dom cdm 4477  [cec 6357   / cqs 6358

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-ec 6361  df-qs 6365

This theorem is referenced by:  elqsn0  6428  ecelqsdm  6429