Theorem qliftfuns 8862

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 2908	. . . . 5 ⊢ Ⅎ𝑦⟨[𝑥]𝑅, 𝐴⟩
3		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑥[𝑦]𝑅
4		nfcsb1v 3946	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
5	3, 4	nfop 4913	. . . . 5 ⊢ Ⅎ𝑥⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩
6		eceq1 8802	. . . . . 6 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7		csbeq1a 3935	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
8	6, 7	opeq12d 4905	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨[𝑥]𝑅, 𝐴⟩ = ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
9	2, 5, 8	cbvmpt 5277	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
10	9	rneqi 5962	. . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = ran (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
11	1, 10	eqtri 2768	. 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
12		qlift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
13	12	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌)
14	4	nfel1 2925	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌
15	7	eleq1d 2829	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑌 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌))
16	14, 15	rspc 3623	. . 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 3924	. 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
21	11, 17, 18, 19, 20	qliftfun 8860	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 2108  ∀wral 3067  ⦋csb 3921  ⟨cop 4654   class class class wbr 5166   ↦ cmpt 5249  ran crn 5701  Fun wfun 6567   Er wer 8760  [cec 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-er 8763  df-ec 8765  df-qs 8769

This theorem is referenced by: (None)