Theorem ecexr 6632

Description: An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
ecexr	⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)

Proof of Theorem ecexr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elimag 5031	. . . . 5 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴))
2	1	ibi 176	. . . 4 ⊢ (𝐴 ∈ (𝑅 “ {𝐵}) → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)
3		df-ec 6629	. . . 4 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵})
4	2, 3	eleq2s 2301	. . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 ∈ {𝐵}𝑥𝑅𝐴)
5		df-rex 2491	. . . 4 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 ↔ ∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴))
6		simpl 109	. . . . . 6 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 ∈ {𝐵})
7		velsn 3651	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
8	6, 7	sylib 122	. . . . 5 ⊢ ((𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → 𝑥 = 𝐵)
9	8	eximi 1624	. . . 4 ⊢ (∃𝑥(𝑥 ∈ {𝐵} ∧ 𝑥𝑅𝐴) → ∃𝑥 𝑥 = 𝐵)
10	5, 9	sylbi 121	. . 3 ⊢ (∃𝑥 ∈ {𝐵}𝑥𝑅𝐴 → ∃𝑥 𝑥 = 𝐵)
11	4, 10	syl 14	. 2 ⊢ (𝐴 ∈ [𝐵]𝑅 → ∃𝑥 𝑥 = 𝐵)
12		isset 2779	. 2 ⊢ (𝐵 ∈ V ↔ ∃𝑥 𝑥 = 𝐵)
13	11, 12	sylibr 134	1 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∃wrex 2486  Vcvv 2773  {csn 3634   class class class wbr 4047   “ cima 4682  [cec 6625

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-xp 4685  df-cnv 4687  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-ec 6629

This theorem is referenced by:  relelec  6669  ecdmn0m  6671