Theorem qliftel 6879

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 6877	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6	1, 5, 2	fliftel 5989	. 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 6844	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶𝑅𝑥 ↔ [𝐶]𝑅 = [𝑥]𝑅))
10	9	anbi1d 469	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶𝑅𝑥 ∧ 𝐷 = 𝐴) ↔ ([𝐶]𝑅 = [𝑥]𝑅 ∧ 𝐷 = 𝐴)))
11	10	rexbidva 2547	. 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 1402   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821  ⟨cop 3708   class class class wbr 4125   ↦ cmpt 4187  ran crn 4770   Er wer 6794  [cec 6795   / cqs 6796

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-er 6797  df-ec 6799  df-qs 6803

This theorem is referenced by: (None)