Theorem fvineqsneq 34696

Description: A theorem about functions where the image of every point intersects the domain only at that point. (Contributed by ML, 28-Mar-2021.)

Assertion

Ref	Expression
fvineqsneq	⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑍 ⊆ ran 𝐹 ∧ 𝐴 ⊆ ∪ 𝑍)) → 𝑍 = ran 𝐹)

Distinct variable groups:   𝐴,𝑝   𝐹,𝑝   𝑍,𝑝

Proof of Theorem fvineqsneq

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pssnel 4420	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 ⊊ ran 𝐹 → ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥 ∈ 𝑍))
2	1	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥 ∈ 𝑍))
3		df-rex 3144	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑥 ∈ ran 𝐹 ¬ 𝑥 ∈ 𝑍 ↔ ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥 ∈ 𝑍))
4	2, 3	sylibr 236	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹 ¬ 𝑥 ∈ 𝑍)
5		fnrnfv 6725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑥 ∣ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)})
6	5	abeq2d 2947	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)))
7	6	biimpd 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 → ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)))
8	7	ralrimiv 3181	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 Fn 𝐴 → ∀𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝))
9	8	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝))
10	9	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝))
11		r19.29r 3255	. . . . . . . . . . . . . . . . . 18 ⊢ ((∃𝑥 ∈ ran 𝐹 ¬ 𝑥 ∈ 𝑍 ∧ ∀𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)) → ∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)))
12	4, 10, 11	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)))
13		nfra1 3219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑝∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}
14		rsp 3205	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}))
15		vsnid 4602	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑝 ∈ {𝑝}
16		eleq2 2901	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 ∈ {𝑝}))
17	15, 16	mpbiri 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → 𝑝 ∈ ((𝐹‘𝑝) ∩ 𝐴))
18	17	elin1d 4175	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → 𝑝 ∈ (𝐹‘𝑝))
19	14, 18	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → 𝑝 ∈ (𝐹‘𝑝)))
20	19	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹‘𝑝)) → (𝑝 ∈ 𝐴 → 𝑝 ∈ (𝐹‘𝑝)))
21		eleq2 2901	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (𝐹‘𝑝) → (𝑝 ∈ 𝑥 ↔ 𝑝 ∈ (𝐹‘𝑝)))
22	21	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹‘𝑝)) → (𝑝 ∈ 𝑥 ↔ 𝑝 ∈ (𝐹‘𝑝)))
23	20, 22	sylibrd 261	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹‘𝑝)) → (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝑥))
24	23	ex 415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑥 = (𝐹‘𝑝) → (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝑥)))
25	24	com23 86	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → (𝑥 = (𝐹‘𝑝) → 𝑝 ∈ 𝑥)))
26	13, 25	reximdai 3311	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝) → ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥))
27	26	adantl 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝) → ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥))
28	27	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝) → ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥))
29	28	anim2d 613	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ((¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)) → (¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥)))
30	29	reximdv 3273	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)) → ∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥)))
31	12, 30	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥))
32		ancom 463	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥) ↔ (∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
33		r19.41v 3347	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑝 ∈ 𝐴 (𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ↔ (∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
34	32, 33	bitr4i 280	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥) ↔ ∃𝑝 ∈ 𝐴 (𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
35	34	rexbii 3247	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥) ↔ ∃𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 (𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
36	31, 35	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 (𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
37		rexcom 3355	. . . . . . . . . . . . . . 15 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ↔ ∃𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 (𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
38	36, 37	sylibr 236	. . . . . . . . . . . . . 14 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
39		nfre1 3306	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)
40	39	19.3 2202	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ↔ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
41		alral 3154	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → ∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
42	40, 41	sylbir 237	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → ∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
43	42	reximi 3243	. . . . . . . . . . . . . 14 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
44	38, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍))
45		nfv 1915	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝 𝐹 Fn 𝐴
46	45, 13	nfan 1900	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝(𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})
47		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝 𝑍 ⊊ ran 𝐹
48	46, 47	nfan 1900	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑝((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹)
49		nfv 1915	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝 ∈ 𝐴)
50		fvineqsneu 34695	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑝 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥)
51	50	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑝 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥)
52		rsp 3205	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑝 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥 → (𝑝 ∈ 𝐴 → ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥))
53	51, 52	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (𝑝 ∈ 𝐴 → ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥))
54	53	adantrd 494	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ((𝑝 ∈ 𝐴 ∧ 𝑥 ∈ ran 𝐹) → ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥))
55	54	imp 409	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝 ∈ 𝐴 ∧ 𝑥 ∈ ran 𝐹)) → ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥)
56		reupick3 4288	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ 𝑥 ∈ ran 𝐹) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))
57	56	3expa 1114	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) ∧ 𝑥 ∈ ran 𝐹) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))
58	57	expcom 416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ran 𝐹 → ((∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))
59	58	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑥 ∈ ran 𝐹) → ((∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))
60	59	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝 ∈ 𝐴 ∧ 𝑥 ∈ ran 𝐹)) → ((∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))
61	55, 60	mpand 693	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝 ∈ 𝐴 ∧ 𝑥 ∈ ran 𝐹)) → (∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))
62	61	expr 459	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝 ∈ 𝐴) → (𝑥 ∈ ran 𝐹 → (∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))))
63	49, 62	ralrimi 3216	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝 ∈ 𝐴) → ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))
64	63	ex 415	. . . . . . . . . . . . . 14 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (𝑝 ∈ 𝐴 → ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))))
65	48, 64	ralrimi 3216	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))
66		r19.29r 3255	. . . . . . . . . . . . 13 ⊢ ((∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ ∀𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) → ∃𝑝 ∈ 𝐴 (∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))))
67	44, 65, 66	syl2anc 586	. . . . . . . . . . . 12 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 (∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))))
68		ralim 3162	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)) → (∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → ∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))
69	68	impcom 410	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) → ∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))
70	69	reximi 3243	. . . . . . . . . . . 12 ⊢ (∃𝑝 ∈ 𝐴 (∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))
71	67, 70	syl 17	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))
72		con2b 362	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍) ↔ (𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥))
73	72	ralbii 3165	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍) ↔ ∀𝑥 ∈ ran 𝐹(𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥))
74		df-ral 3143	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ ran 𝐹(𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ ran 𝐹 → (𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥)))
75		bi2.04 391	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ran 𝐹 → (𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥)) ↔ (𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)))
76	75	albii 1820	. . . . . . . . . . . . . 14 ⊢ (∀𝑥(𝑥 ∈ ran 𝐹 → (𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥)) ↔ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)))
77	73, 74, 76	3bitri 299	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍) ↔ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)))
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍) ↔ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))))
79	48, 78	rexbid 3320	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍) ↔ ∃𝑝 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))))
80	71, 79	mpbid 234	. . . . . . . . . 10 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)))
81		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹)
82		nfa1 2155	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))
83	81, 82	nfan 1900	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)))
84		pssss 4072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 ⊊ ran 𝐹 → 𝑍 ⊆ ran 𝐹)
85		dfss2 3955	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑍 ⊆ ran 𝐹 ↔ ∀𝑥(𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹))
86	84, 85	sylib 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 ⊊ ran 𝐹 → ∀𝑥(𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹))
87	86	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥(𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹))
88		df-ral 3143	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ 𝑍 𝑥 ∈ ran 𝐹 ↔ ∀𝑥(𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹))
89	87, 88	sylibr 236	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥 ∈ 𝑍 𝑥 ∈ ran 𝐹)
90	89	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → ∀𝑥 ∈ 𝑍 𝑥 ∈ ran 𝐹)
91		rsp 3205	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑍 𝑥 ∈ ran 𝐹 → (𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹))
92	90, 91	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → (𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹))
93		df-ral 3143	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ 𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)))
94	93	biimpri 230	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) → ∀𝑥 ∈ 𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))
95	94	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → ∀𝑥 ∈ 𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))
96		rsp 3205	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥) → (𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)))
97	95, 96	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → (𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)))
98	92, 97	mpdd 43	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → (𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥))
99	83, 98	ralrimi 3216	. . . . . . . . . . . . 13 ⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥)
100	99	ex 415	. . . . . . . . . . . 12 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) → ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥))
101	100	a1d 25	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (𝑝 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) → ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥)))
102	48, 101	reximdai 3311	. . . . . . . . . 10 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥))
103	80, 102	mpd 15	. . . . . . . . 9 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥)
104		ralnex 3236	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝑍 𝑝 ∈ 𝑥)
105	104	rexbii 3247	. . . . . . . . 9 ⊢ (∃𝑝 ∈ 𝐴 ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥 ↔ ∃𝑝 ∈ 𝐴 ¬ ∃𝑥 ∈ 𝑍 𝑝 ∈ 𝑥)
106	103, 105	sylib 220	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ¬ ∃𝑥 ∈ 𝑍 𝑝 ∈ 𝑥)
107		eluni2 4842	. . . . . . . . . 10 ⊢ (𝑝 ∈ ∪ 𝑍 ↔ ∃𝑥 ∈ 𝑍 𝑝 ∈ 𝑥)
108	107	notbii 322	. . . . . . . . 9 ⊢ (¬ 𝑝 ∈ ∪ 𝑍 ↔ ¬ ∃𝑥 ∈ 𝑍 𝑝 ∈ 𝑥)
109	108	rexbii 3247	. . . . . . . 8 ⊢ (∃𝑝 ∈ 𝐴 ¬ 𝑝 ∈ ∪ 𝑍 ↔ ∃𝑝 ∈ 𝐴 ¬ ∃𝑥 ∈ 𝑍 𝑝 ∈ 𝑥)
110	106, 109	sylibr 236	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ¬ 𝑝 ∈ ∪ 𝑍)
111		dfss3 3956	. . . . . . . . 9 ⊢ (𝐴 ⊆ ∪ 𝑍 ↔ ∀𝑝 ∈ 𝐴 𝑝 ∈ ∪ 𝑍)
112		dfral2 3237	. . . . . . . . 9 ⊢ (∀𝑝 ∈ 𝐴 𝑝 ∈ ∪ 𝑍 ↔ ¬ ∃𝑝 ∈ 𝐴 ¬ 𝑝 ∈ ∪ 𝑍)
113	111, 112	bitri 277	. . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝑍 ↔ ¬ ∃𝑝 ∈ 𝐴 ¬ 𝑝 ∈ ∪ 𝑍)
114	113	con2bii2 34617	. . . . . . 7 ⊢ (¬ 𝐴 ⊆ ∪ 𝑍 ↔ ∃𝑝 ∈ 𝐴 ¬ 𝑝 ∈ ∪ 𝑍)
115	110, 114	sylibr 236	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ¬ 𝐴 ⊆ ∪ 𝑍)
116	115	ex 415	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑍 ⊊ ran 𝐹 → ¬ 𝐴 ⊆ ∪ 𝑍))
117	116	con2d 136	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝐴 ⊆ ∪ 𝑍 → ¬ 𝑍 ⊊ ran 𝐹))
118		npss 4087	. . . 4 ⊢ (¬ 𝑍 ⊊ ran 𝐹 ↔ (𝑍 ⊆ ran 𝐹 → 𝑍 = ran 𝐹))
119	117, 118	syl6ib 253	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝐴 ⊆ ∪ 𝑍 → (𝑍 ⊆ ran 𝐹 → 𝑍 = ran 𝐹)))
120	119	com23 86	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑍 ⊆ ran 𝐹 → (𝐴 ⊆ ∪ 𝑍 → 𝑍 = ran 𝐹)))
121	120	imp32 421	1 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑍 ⊆ ran 𝐹 ∧ 𝐴 ⊆ ∪ 𝑍)) → 𝑍 = ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140   ∩ cin 3935   ⊆ wss 3936   ⊊ wpss 3937  {csn 4567  ∪ cuni 4838  ran crn 5556   Fn wfn 6350  ‘cfv 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fv 6363

This theorem is referenced by:  pibt2  34701