Theorem ptcmplem4 22138

Description: Lemma for ptcmp 22141. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypotheses

Ref	Expression
ptcmp.1	⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
ptcmp.2	⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
ptcmp.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ptcmp.4	⊢ (𝜑 → 𝐹:𝐴⟶Comp)
ptcmp.5	⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
ptcmplem2.5	⊢ (𝜑 → 𝑈 ⊆ ran 𝑆)
ptcmplem2.6	⊢ (𝜑 → 𝑋 = ∪ 𝑈)
ptcmplem2.7	⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)
ptcmplem3.8	⊢ 𝐾 = {𝑢 ∈ (𝐹‘𝑘) ∣ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑈}

Assertion

Ref	Expression
ptcmplem4	⊢ ¬ 𝜑

Distinct variable groups:   𝑘,𝑛,𝑢,𝑤,𝑧,𝐴   𝑢,𝐾   𝑆,𝑘,𝑛,𝑢,𝑧   𝜑,𝑘,𝑛,𝑢   𝑈,𝑘,𝑢,𝑧   𝑘,𝑉,𝑛,𝑢,𝑤,𝑧   𝑘,𝐹,𝑛,𝑢,𝑤,𝑧   𝑘,𝑋,𝑛,𝑢,𝑤,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝑆(𝑤)   𝑈(𝑤,𝑛)   𝐾(𝑧,𝑤,𝑘,𝑛)

Proof of Theorem ptcmplem4

Dummy variables 𝑓 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptcmp.1	. . 3 ⊢ 𝑆 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
2		ptcmp.2	. . 3 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
3		ptcmp.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		ptcmp.4	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶Comp)
5		ptcmp.5	. . 3 ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
6		ptcmplem2.5	. . 3 ⊢ (𝜑 → 𝑈 ⊆ ran 𝑆)
7		ptcmplem2.6	. . 3 ⊢ (𝜑 → 𝑋 = ∪ 𝑈)
8		ptcmplem2.7	. . 3 ⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)
9		ptcmplem3.8	. . 3 ⊢ 𝐾 = {𝑢 ∈ (𝐹‘𝑘) ∣ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑈}
10	1, 2, 3, 4, 5, 6, 7, 8, 9	ptcmplem3 22137	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
11		simprl 787	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 Fn 𝐴)
12		eldifi 3894	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
13	12	ralimi 3099	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
14		fveq2 6375	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
15		fveq2 6375	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
16	15	unieqd 4604	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑘))
17	14, 16	eleq12d 2838	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛) ↔ (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘)))
18	17	cbvralv 3319	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛) ↔ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
19	13, 18	sylibr 225	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
20	19	ad2antll 720	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
21		vex 3353	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
22	21	elixp 8120	. . . . . . . . 9 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛)))
23	11, 20, 22	sylanbrc 578	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
24	23, 2	syl6eleqr 2855	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ 𝑋)
25	7	adantr 472	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑋 = ∪ 𝑈)
26	24, 25	eleqtrd 2846	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ ∪ 𝑈)
27		eluni2 4598	. . . . . 6 ⊢ (𝑓 ∈ ∪ 𝑈 ↔ ∃𝑣 ∈ 𝑈 𝑓 ∈ 𝑣)
28	26, 27	sylib 209	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∃𝑣 ∈ 𝑈 𝑓 ∈ 𝑣)
29		simplrr 796	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ 𝑣)
30	29	adantr 472	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑢 ∈ (𝐹‘𝑘) ∧ 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → 𝑓 ∈ 𝑣)
31		simprr 789	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑢 ∈ (𝐹‘𝑘) ∧ 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
32	30, 31	eleqtrd 2846	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑢 ∈ (𝐹‘𝑘) ∧ 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → 𝑓 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
33		fveq1 6374	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑓 → (𝑤‘𝑘) = (𝑓‘𝑘))
34	33	eleq1d 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑓 → ((𝑤‘𝑘) ∈ 𝑢 ↔ (𝑓‘𝑘) ∈ 𝑢))
35		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
36	35	mptpreima 5814	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝑋 ∣ (𝑤‘𝑘) ∈ 𝑢}
37	34, 36	elrab2 3523	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ 𝑋 ∧ (𝑓‘𝑘) ∈ 𝑢))
38	37	simprbi 490	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑓‘𝑘) ∈ 𝑢)
39	32, 38	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑢 ∈ (𝐹‘𝑘) ∧ 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → (𝑓‘𝑘) ∈ 𝑢)
40		simprl 787	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑢 ∈ (𝐹‘𝑘) ∧ 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → 𝑢 ∈ (𝐹‘𝑘))
41		simplrl 795	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑣 ∈ 𝑈)
42	41	adantr 472	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑢 ∈ (𝐹‘𝑘) ∧ 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → 𝑣 ∈ 𝑈)
43	31, 42	eqeltrrd 2845	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑢 ∈ (𝐹‘𝑘) ∧ 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑈)
44		rabid 3263	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ {𝑢 ∈ (𝐹‘𝑘) ∣ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑈} ↔ (𝑢 ∈ (𝐹‘𝑘) ∧ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑈))
45	40, 43, 44	sylanbrc 578	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑢 ∈ (𝐹‘𝑘) ∧ 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → 𝑢 ∈ {𝑢 ∈ (𝐹‘𝑘) ∣ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ 𝑈})
46	45, 9	syl6eleqr 2855	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑢 ∈ (𝐹‘𝑘) ∧ 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → 𝑢 ∈ 𝐾)
47		elunii 4599	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑘) ∈ 𝑢 ∧ 𝑢 ∈ 𝐾) → (𝑓‘𝑘) ∈ ∪ 𝐾)
48	39, 46, 47	syl2anc 579	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑢 ∈ (𝐹‘𝑘) ∧ 𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))) → (𝑓‘𝑘) ∈ ∪ 𝐾)
49	48	rexlimdvaa 3179	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → (∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑓‘𝑘) ∈ ∪ 𝐾))
50	49	expr 448	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ 𝑘 ∈ 𝐴) → ((𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → (∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑓‘𝑘) ∈ ∪ 𝐾)))
51	50	ralimdva 3109	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) → (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑘 ∈ 𝐴 (∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑓‘𝑘) ∈ ∪ 𝐾)))
52	51	ex 401	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 Fn 𝐴) → ((𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣) → (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑘 ∈ 𝐴 (∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑓‘𝑘) ∈ ∪ 𝐾))))
53	52	com23 86	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 Fn 𝐴) → (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ((𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣) → ∀𝑘 ∈ 𝐴 (∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑓‘𝑘) ∈ ∪ 𝐾))))
54	53	impr 446	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ((𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣) → ∀𝑘 ∈ 𝐴 (∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑓‘𝑘) ∈ ∪ 𝐾)))
55	54	imp 395	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) → ∀𝑘 ∈ 𝐴 (∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑓‘𝑘) ∈ ∪ 𝐾))
56	6	adantr 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑈 ⊆ ran 𝑆)
57	56	sselda 3761	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ ran 𝑆)
58	57	adantrr 708	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) → 𝑣 ∈ ran 𝑆)
59	1	rnmpt2 6968	. . . . . . . 8 ⊢ ran 𝑆 = {𝑣 ∣ ∃𝑘 ∈ 𝐴 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)}
60	58, 59	syl6eleq 2854	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) → 𝑣 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐴 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)})
61		abid 2753	. . . . . . 7 ⊢ (𝑣 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐴 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)} ↔ ∃𝑘 ∈ 𝐴 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
62	60, 61	sylib 209	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) → ∃𝑘 ∈ 𝐴 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
63		rexim 3154	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 (∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑓‘𝑘) ∈ ∪ 𝐾) → (∃𝑘 ∈ 𝐴 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾))
64	55, 62, 63	sylc 65	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) → ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾)
65	28, 64	rexlimddv 3182	. . . 4 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾)
66		eldifn 3895	. . . . . . 7 ⊢ ((𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
67	66	ralimi 3099	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
68	67	ad2antll 720	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
69		ralnex 3139	. . . . 5 ⊢ (∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾 ↔ ¬ ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾)
70	68, 69	sylib 209	. . . 4 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ¬ ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾)
71	65, 70	pm2.65da 851	. . 3 ⊢ (𝜑 → ¬ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
72	71	nexdv 2031	. 2 ⊢ (𝜑 → ¬ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
73	10, 72	pm2.65i 185	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1652  ∃wex 1874   ∈ wcel 2155  {cab 2751  ∀wral 3055  ∃wrex 3056  {crab 3059   ∖ cdif 3729   ∩ cin 3731   ⊆ wss 3732  𝒫 cpw 4315  ∪ cuni 4594   ↦ cmpt 4888  ^◡ccnv 5276  dom cdm 5277  ran crn 5278   “ cima 5280   Fn wfn 6063  ⟶wf 6064  ‘cfv 6068   ↦ cmpt2 6844  Xcixp 8113  Fincfn 8160  cardccrd 9012  Compccmp 21469  UFLcufl 21983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-omul 7769  df-er 7947  df-map 8062  df-ixp 8114  df-en 8161  df-dom 8162  df-fin 8164  df-wdom 8671  df-card 9016  df-acn 9019  df-cmp 21470

This theorem is referenced by:  ptcmplem5  22139