Theorem ecss 6083

Description: An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
ecss.1	⊢ (φ → 𝑅 Er 𝑋)

Assertion

Ref	Expression
ecss	⊢ (φ → [A]𝑅 ⊆ 𝑋)

Proof of Theorem ecss

Step	Hyp	Ref	Expression
1		df-ec 6044	. . 3 ⊢ [A]𝑅 = (𝑅 “ {A})
2		imassrn 4622	. . 3 ⊢ (𝑅 “ {A}) ⊆ ran 𝑅
3	1, 2	eqsstri 2969	. 2 ⊢ [A]𝑅 ⊆ ran 𝑅
4		ecss.1	. . 3 ⊢ (φ → 𝑅 Er 𝑋)
5		errn 6064	. . 3 ⊢ (𝑅 Er 𝑋 → ran 𝑅 = 𝑋)
6	4, 5	syl 14	. 2 ⊢ (φ → ran 𝑅 = 𝑋)
7	3, 6	syl5sseq 2987	1 ⊢ (φ → [A]𝑅 ⊆ 𝑋)

Syntax hints:   → wi 4   = wceq 1242   ⊆ wss 2911  {csn 3367  ran crn 4289   “ cima 4291   Er wer 6039  [cec 6040

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-er 6042  df-ec 6044

This theorem is referenced by:  qsss  6101