Theorem fliftfun 7321

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)
fliftfun.4	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐶)
fliftfun.5	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐷)

Assertion

Ref	Expression
fliftfun	⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷)))

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

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑦)   𝐹(𝑥)

Proof of Theorem fliftfun

Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1947	. . 3 ⊢ Ⅎ𝑥𝜑
2		flift.1	. . . . 5 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
3		nfmpt1 5215	. . . . . 6 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
4	3	nfrn 5947	. . . . 5 ⊢ Ⅎ𝑥ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
5	2, 4	nfcxfr 2926	. . . 4 ⊢ Ⅎ𝑥𝐹
6	5	nffun 6566	. . 3 ⊢ Ⅎ𝑥Fun 𝐹
7		fveq2 6888	. . . . . . 7 ⊢ (𝐴 = 𝐶 → (𝐹‘𝐴) = (𝐹‘𝐶))
8		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → Fun 𝐹)
9		flift.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
10		flift.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
11	2, 9, 10	fliftel1 7319	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴𝐹𝐵)
12	11	ad2ant2r 760	. . . . . . . . 9 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐴𝐹𝐵)
13		funbrfv 6936	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵))
14	8, 12, 13	sylc 66	. . . . . . . 8 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝐴) = 𝐵)
15		simprr 785	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
16		eqidd 2767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐶 = 𝐶)
17		eqidd 2767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐷 = 𝐷)
18		fliftfun.4	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐶)
19	18	eqeq2d 2777	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐶))
20		fliftfun.5	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐷)
21	20	eqeq2d 2777	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐷 = 𝐵 ↔ 𝐷 = 𝐷))
22	19, 21	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)))
23	22	rspcev 3584	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑋 ∧ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)) → ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
24	15, 16, 17, 23	syl12anc 850	. . . . . . . . . 10 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
25	2, 9, 10	fliftel 7318	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
26	25	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
27	24, 26	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐶𝐹𝐷)
28		funbrfv 6936	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹‘𝐶) = 𝐷))
29	8, 27, 28	sylc 66	. . . . . . . 8 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝐶) = 𝐷)
30	14, 29	eqeq12d 2782	. . . . . . 7 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝐴) = (𝐹‘𝐶) ↔ 𝐵 = 𝐷))
31	7, 30	imbitrid 247	. . . . . 6 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐴 = 𝐶 → 𝐵 = 𝐷))
32	31	anassrs 473	. . . . 5 ⊢ ((((𝜑 ∧ Fun 𝐹) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐴 = 𝐶 → 𝐵 = 𝐷))
33	32	ralrimiva 3160	. . . 4 ⊢ (((𝜑 ∧ Fun 𝐹) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷))
34	33	exp31 425	. . 3 ⊢ (𝜑 → (Fun 𝐹 → (𝑥 ∈ 𝑋 → ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷))))
35	1, 6, 34	ralrimd 3273	. 2 ⊢ (𝜑 → (Fun 𝐹 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷)))
36	2, 9, 10	fliftel 7318	. . . . . . . . 9 ⊢ (𝜑 → (𝑧𝐹𝑢 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑢 = 𝐵)))
37	2, 9, 10	fliftel 7318	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧𝐹𝑣 ↔ ∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑣 = 𝐵)))
38	18	eqeq2d 2777	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐴 ↔ 𝑧 = 𝐶))
39	20	eqeq2d 2777	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑣 = 𝐵 ↔ 𝑣 = 𝐷))
40	38, 39	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝐴 ∧ 𝑣 = 𝐵) ↔ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷)))
41	40	cbvrexvw 3247	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑣 = 𝐵) ↔ ∃𝑦 ∈ 𝑋 (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))
42	37, 41	bitrdi 290	. . . . . . . . 9 ⊢ (𝜑 → (𝑧𝐹𝑣 ↔ ∃𝑦 ∈ 𝑋 (𝑧 = 𝐶 ∧ 𝑣 = 𝐷)))
43	36, 42	anbi12d 644	. . . . . . . 8 ⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑧𝐹𝑣) ↔ (∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ ∃𝑦 ∈ 𝑋 (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))))
44	43	biimpd 232	. . . . . . 7 ⊢ (𝜑 → ((𝑧𝐹𝑢 ∧ 𝑧𝐹𝑣) → (∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ ∃𝑦 ∈ 𝑋 (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))))
45		reeanv 3240	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷)) ↔ (∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ ∃𝑦 ∈ 𝑋 (𝑧 = 𝐶 ∧ 𝑣 = 𝐷)))
46		r19.29 3131	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))) → ∃𝑥 ∈ 𝑋 (∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ∃𝑦 ∈ 𝑋 ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))))
47		r19.29 3131	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ∃𝑦 ∈ 𝑋 ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))) → ∃𝑦 ∈ 𝑋 ((𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))))
48		eqtr2 2787	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 𝐴 ∧ 𝑧 = 𝐶) → 𝐴 = 𝐶)
49	48	ad2ant2r 760	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷)) → 𝐴 = 𝐶)
50	49	imim1i 64	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 = 𝐶 → 𝐵 = 𝐷) → (((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷)) → 𝐵 = 𝐷))
51	50	imp 412	. . . . . . . . . . . . . 14 ⊢ (((𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))) → 𝐵 = 𝐷)
52		simprlr 792	. . . . . . . . . . . . . 14 ⊢ (((𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))) → 𝑢 = 𝐵)
53		simprrr 794	. . . . . . . . . . . . . 14 ⊢ (((𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))) → 𝑣 = 𝐷)
54	51, 52, 53	3eqtr4d 2811	. . . . . . . . . . . . 13 ⊢ (((𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))) → 𝑢 = 𝑣)
55	54	rexlimivw 3165	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝑋 ((𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))) → 𝑢 = 𝑣)
56	47, 55	syl 18	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ∃𝑦 ∈ 𝑋 ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))) → 𝑢 = 𝑣)
57	56	rexlimivw 3165	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝑋 (∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ∃𝑦 ∈ 𝑋 ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))) → 𝑢 = 𝑣)
58	46, 57	syl 18	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) ∧ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷))) → 𝑢 = 𝑣)
59	58	ex 418	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) → (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑣 = 𝐷)) → 𝑢 = 𝑣))
60	45, 59	biimtrrid 246	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) → ((∃𝑥 ∈ 𝑋 (𝑧 = 𝐴 ∧ 𝑢 = 𝐵) ∧ ∃𝑦 ∈ 𝑋 (𝑧 = 𝐶 ∧ 𝑣 = 𝐷)) → 𝑢 = 𝑣))
61	44, 60	syl9 78	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) → ((𝑧𝐹𝑢 ∧ 𝑧𝐹𝑣) → 𝑢 = 𝑣)))
62	61	alrimdv 1962	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) → ∀𝑣((𝑧𝐹𝑢 ∧ 𝑧𝐹𝑣) → 𝑢 = 𝑣)))
63	62	alrimdv 1962	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) → ∀𝑢∀𝑣((𝑧𝐹𝑢 ∧ 𝑧𝐹𝑣) → 𝑢 = 𝑣)))
64	63	alrimdv 1962	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) → ∀𝑧∀𝑢∀𝑣((𝑧𝐹𝑢 ∧ 𝑧𝐹𝑣) → 𝑢 = 𝑣)))
65	2, 9, 10	fliftrel 7317	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆))
66		relxp 5684	. . . . 5 ⊢ Rel (𝑅 × 𝑆)
67		relss 5773	. . . . 5 ⊢ (𝐹 ⊆ (𝑅 × 𝑆) → (Rel (𝑅 × 𝑆) → Rel 𝐹))
68	65, 66, 67	mpisyl 22	. . . 4 ⊢ (𝜑 → Rel 𝐹)
69		dffun2 6553	. . . . 5 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑧∀𝑢∀𝑣((𝑧𝐹𝑢 ∧ 𝑧𝐹𝑣) → 𝑢 = 𝑣)))
70	69	baib 545	. . . 4 ⊢ (Rel 𝐹 → (Fun 𝐹 ↔ ∀𝑧∀𝑢∀𝑣((𝑧𝐹𝑢 ∧ 𝑧𝐹𝑣) → 𝑢 = 𝑣)))
71	68, 70	syl 18	. . 3 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑧∀𝑢∀𝑣((𝑧𝐹𝑢 ∧ 𝑧𝐹𝑣) → 𝑢 = 𝑣)))
72	64, 71	sylibrd 262	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷) → Fun 𝐹))
73	35, 72	impbid 215	1 ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  ∀wral 3082  ∃wrex 3092   ⊆ wss 3908  ⟨cop 4600   class class class wbr 5114   ↦ cmpt 5197   × cxp 5664  ran crn 5667  Rel wrel 5671  Fun wfun 6537  ‘cfv 6543

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551

This theorem is used by:  fliftfund  7322  fliftfuns  7323  qliftfun  8809