Theorem fliftfuns 5692

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 2279	. . . . 5 ⊢ Ⅎ𝑦⟨𝐴, 𝐵⟩
3		nfcsb1v 3030	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
4		nfcsb1v 3030	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5	3, 4	nfop 3716	. . . . 5 ⊢ Ⅎ𝑥⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩
6		csbeq1a 3007	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
7		csbeq1a 3007	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
8	6, 7	opeq12d 3708	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
9	2, 5, 8	cbvmpt 4018	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑦 ∈ 𝑋 ↦ ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
10	9	rneqi 4762	. . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = ran (𝑦 ∈ 𝑋 ↦ ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
11	1, 10	eqtri 2158	. 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
12		flift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
13	12	ralrimiva 2503	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑅)
14	3	nfel1 2290	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅
15	6	eleq1d 2206	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑅 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅))
16	14, 15	rspc 2778	. . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑅 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅))
17	13, 16	mpan9 279	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅)
18		flift.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
19	18	ralrimiva 2503	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆)
20	4	nfel1 2290	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆
21	7	eleq1d 2206	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑆 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆))
22	20, 21	rspc 2778	. . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆 → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆))
23	19, 22	mpan9 279	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆)
24		csbeq1 3001	. 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
25		csbeq1 3001	. 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
26	11, 17, 23, 24, 25	fliftfun 5690	1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2414  ⦋csb 2998  ⟨cop 3525   ↦ cmpt 3984  ran crn 4535  Fun wfun 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126

This theorem is referenced by: (None)