Theorem elqsn0m 6815

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 2231	. 2 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅)
2		eleq2 2295	. . 3 ⊢ ([𝑦]𝑅 = 𝐵 → (𝑥 ∈ [𝑦]𝑅 ↔ 𝑥 ∈ 𝐵))
3	2	exbidv 1873	. 2 ⊢ ([𝑦]𝑅 = 𝐵 → (∃𝑥 𝑥 ∈ [𝑦]𝑅 ↔ ∃𝑥 𝑥 ∈ 𝐵))
4		eleq2 2295	. . . 4 ⊢ (dom 𝑅 = 𝐴 → (𝑦 ∈ dom 𝑅 ↔ 𝑦 ∈ 𝐴))
5	4	biimpar 297	. . 3 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ dom 𝑅)
6		ecdmn0m 6789	. . 3 ⊢ (𝑦 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝑦]𝑅)
7	5, 6	sylib 122	. 2 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 𝑥 ∈ [𝑦]𝑅)
8	1, 3, 7	ectocld 6813	1 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202  dom cdm 4731  [cec 6743   / cqs 6744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-ec 6747  df-qs 6751

This theorem is referenced by:  elqsn0  6816  ecelqsdm  6817