Theorem fliftfuns 5938

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 2374	. . . . 5 ⊢ Ⅎ𝑦⟨𝐴, 𝐵⟩
3		nfcsb1v 3160	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
4		nfcsb1v 3160	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5	3, 4	nfop 3878	. . . . 5 ⊢ Ⅎ𝑥⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩
6		csbeq1a 3136	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
7		csbeq1a 3136	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
8	6, 7	opeq12d 3870	. . . . 5 ⊢ (𝑥 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
9	2, 5, 8	cbvmpt 4184	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑦 ∈ 𝑋 ↦ ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
10	9	rneqi 4960	. . 3 ⊢ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = ran (𝑦 ∈ 𝑋 ↦ ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
11	1, 10	eqtri 2252	. 2 ⊢ 𝐹 = ran (𝑦 ∈ 𝑋 ↦ ⟨⦋𝑦 / 𝑥⦌𝐴, ⦋𝑦 / 𝑥⦌𝐵⟩)
12		flift.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
13	12	ralrimiva 2605	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑅)
14	3	nfel1 2385	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅
15	6	eleq1d 2300	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑅 ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅))
16	14, 15	rspc 2904	. . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐴 ∈ 𝑅 → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅))
17	13, 16	mpan9 281	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐴 ∈ 𝑅)
18		flift.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
19	18	ralrimiva 2605	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆)
20	4	nfel1 2385	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆
21	7	eleq1d 2300	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ 𝑆 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆))
22	20, 21	rspc 2904	. . 3 ⊢ (𝑦 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆 → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆))
23	19, 22	mpan9 281	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝑆)
24		csbeq1 3130	. 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
25		csbeq1 3130	. 2 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
26	11, 17, 23, 24, 25	fliftfun 5936	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 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ⦋csb 3127  ⟨cop 3672   ↦ cmpt 4150  ran crn 4726  Fun wfun 5320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334

This theorem is referenced by: (None)