Theorem qliftel 6779

Description: Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
qlift.1	⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
qlift.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
qlift.3	⊢ (𝜑 → 𝑅 Er 𝑋)
qlift.4	⊢ (𝜑 → 𝑋 ∈ V)

Assertion

Ref	Expression
qliftel	⊢ (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶𝑅𝑥 ∧ 𝐷 = 𝐴)))

Distinct variable groups:   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋   𝑥,𝑌

Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem qliftel

Step	Hyp	Ref	Expression
1		qlift.1	. . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2		qlift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
3		qlift.3	. . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋)
4		qlift.4	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
5	1, 2, 3, 4	qliftlem 6777	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6	1, 5, 2	fliftel 5929	. 2 ⊢ (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 ([𝐶]𝑅 = [𝑥]𝑅 ∧ 𝐷 = 𝐴)))
7	3	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑅 Er 𝑋)
8		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
9	7, 8	erth2 6744	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶𝑅𝑥 ↔ [𝐶]𝑅 = [𝑥]𝑅))
10	9	anbi1d 465	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶𝑅𝑥 ∧ 𝐷 = 𝐴) ↔ ([𝐶]𝑅 = [𝑥]𝑅 ∧ 𝐷 = 𝐴)))
11	10	rexbidva 2527	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 (𝐶𝑅𝑥 ∧ 𝐷 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 ([𝐶]𝑅 = [𝑥]𝑅 ∧ 𝐷 = 𝐴)))
12	6, 11	bitr4d 191	1 ⊢ (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶𝑅𝑥 ∧ 𝐷 = 𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  Vcvv 2800  ⟨cop 3670   class class class wbr 4086   ↦ cmpt 4148  ran crn 4724   Er wer 6694  [cec 6695   / cqs 6696

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-er 6697  df-ec 6699  df-qs 6703

This theorem is referenced by: (None)