Theorem qliftfuns 8843

Description: The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)

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
qliftfuns	⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)))

Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑦,𝑧,𝜑   𝑥,𝑅,𝑦,𝑧   𝑦,𝐹,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem qliftfuns

Step	Hyp	Ref	Expression
1		qlift.1	. . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑦⟨[𝑥]𝑅, 𝐴⟩
3		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑥[𝑦]𝑅
4		nfcsb1v 3933	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
5	3, 4	nfop 4894	. . . . 5 ⊢ Ⅎ𝑥⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩
6		eceq1 8783	. . . . . 6 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7		csbeq1a 3922	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
8	6, 7	opeq12d 4886	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨[𝑥]𝑅, 𝐴⟩ = ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
9	2, 5, 8	cbvmpt 5259	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
10	9	rneqi 5951	. . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = ran (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
11	1, 10	eqtri 2763	. 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
12		qlift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
13	12	ralrimiva 3144	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌)
14	4	nfel1 2920	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌
15	7	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑌 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌))
16	14, 15	rspc 3610	. . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌))
17	13, 16	mpan9 506	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌)
18		qlift.3	. 2 ⊢ (𝜑 → 𝑅 Er 𝑋)
19		qlift.4	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
20		csbeq1 3911	. 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
21	11, 17, 18, 19, 20	qliftfun 8841	1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ⦋csb 3908  ⟨cop 4637   class class class wbr 5148   ↦ cmpt 5231  ran crn 5690  Fun wfun 6557   Er wer 8741  [cec 8742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-er 8744  df-ec 8746  df-qs 8750

This theorem is referenced by: (None)