Theorem qliftfuns 8744

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 2899	. . . . 5 ⊢ Ⅎ𝑦⟨[𝑥]𝑅, 𝐴⟩
3		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥[𝑦]𝑅
4		nfcsb1v 3862	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
5	3, 4	nfop 4833	. . . . 5 ⊢ Ⅎ𝑥⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩
6		eceq1 8676	. . . . . 6 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7		csbeq1a 3852	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
8	6, 7	opeq12d 4825	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨[𝑥]𝑅, 𝐴⟩ = ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
9	2, 5, 8	cbvmpt 5188	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
10	9	rneqi 5886	. . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = ran (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
11	1, 10	eqtri 2760	. 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
12		qlift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
13	12	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌)
14	4	nfel1 2916	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌
15	7	eleq1d 2822	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑌 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌))
16	14, 15	rspc 3553	. . 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 3841	. 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
21	11, 17, 18, 19, 20	qliftfun 8742	1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⦋csb 3838  ⟨cop 4574   class class class wbr 5086   ↦ cmpt 5167  ran crn 5625  Fun wfun 6486   Er wer 8633  [cec 8634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-er 8636  df-ec 8638  df-qs 8642

This theorem is referenced by: (None)