Theorem fliftfuns 7319

Description: The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
flift.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
flift.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)

Assertion

Ref	Expression
fliftfuns	⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))

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

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftfuns

Step	Hyp	Ref	Expression
1		flift.1	. . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
2		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑦⟨𝐴, 𝐵⟩
3		nfcsb1v 3874	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
4		nfcsb1v 3874	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5	3, 4	nfop 4852	. . . . 5 ⊢ Ⅎ𝑥⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩
6		csbeq1a 3864	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
7		csbeq1a 3864	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
8	6, 7	opeq12d 4844	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
9	2, 5, 8	cbvmpt 5211	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑦 ∈ 𝑋 ↦ ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
10	9	rneqi 5925	. . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = ran (𝑦 ∈ 𝑋 ↦ ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
11	1, 10	eqtri 2785	. 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
12		flift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
13	12	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑅)
14	3	nfel1 2940	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅
15	6	eleq1d 2847	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑅 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅))
16	14, 15	rspc 3567	. . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑅 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅))
17	13, 16	mpan9 516	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅)
18		flift.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
19	18	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆)
20	4	nfel1 2940	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆
21	7	eleq1d 2847	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑆 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆))
22	20, 21	rspc 3567	. . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆 → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆))
23	19, 22	mpan9 516	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆)
24		csbeq1 3853	. 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
25		csbeq1 3853	. 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
26	11, 17, 23, 24, 25	fliftfun 7317	1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ⦋csb 3850  ⟨cop 4593   ↦ cmpt 5190  ran crn 5660  Fun wfun 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545

This theorem is used by: (None)