Theorem qliftfuns 8781

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 2923	. . . . 5 ⊢ Ⅎ𝑦⟨[𝑥]𝑅, 𝐴⟩
3		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥[𝑦]𝑅
4		nfcsb1v 3876	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
5	3, 4	nfop 4846	. . . . 5 ⊢ Ⅎ𝑥⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩
6		eceq1 8713	. . . . . 6 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7		csbeq1a 3866	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
8	6, 7	opeq12d 4838	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨[𝑥]𝑅, 𝐴⟩ = ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
9	2, 5, 8	cbvmpt 5201	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
10	9	rneqi 5911	. . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = ran (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
11	1, 10	eqtri 2784	. 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ ⟨[𝑦]𝑅, ⦋𝑦 / 𝑥⦌𝐴⟩)
12		qlift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
13	12	ralrimiva 3153	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌)
14	4	nfel1 2939	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌
15	7	eleq1d 2846	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑌 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌))
16	14, 15	rspc 3569	. . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑌 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑌))
17	13, 16	mpan9 514	. 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 3855	. 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
21	11, 17, 18, 19, 20	qliftfun 8779	1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ⦋csb 3852  ⟨cop 4587   class class class wbr 5099   ↦ cmpt 5180  ran crn 5646  Fun wfun 6511   Er wer 8670  [cec 8671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-er 8673  df-ec 8675  df-qs 8679

This theorem is referenced by: (None)