Theorem elqsn0m 6605

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 2177	. 2 ⊢ (𝐴 / 𝑅) = (𝐴 / 𝑅)
2		eleq2 2241	. . 3 ⊢ ([𝑦]𝑅 = 𝐵 → (𝑥 ∈ [𝑦]𝑅 ↔ 𝑥 ∈ 𝐵))
3	2	exbidv 1825	. 2 ⊢ ([𝑦]𝑅 = 𝐵 → (∃𝑥 𝑥 ∈ [𝑦]𝑅 ↔ ∃𝑥 𝑥 ∈ 𝐵))
4		eleq2 2241	. . . 4 ⊢ (dom 𝑅 = 𝐴 → (𝑦 ∈ dom 𝑅 ↔ 𝑦 ∈ 𝐴))
5	4	biimpar 297	. . 3 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ dom 𝑅)
6		ecdmn0m 6579	. . 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 6603	1 ⊢ ((dom 𝑅 = 𝐴 ∧ 𝐵 ∈ (𝐴 / 𝑅)) → ∃𝑥 𝑥 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353  ∃wex 1492   ∈ wcel 2148  dom cdm 4628  [cec 6535   / cqs 6536

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-ec 6539  df-qs 6543

This theorem is referenced by:  elqsn0  6606  ecelqsdm  6607