Theorem ptcmplem4 23312

Description: Lemma for ptcmp 23315. (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 23311	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
11		simprl 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 Fn 𝐴)
12		eldifi 4078	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
13	12	ralimi 3083	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
14		fveq2 6830	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
15		fveq2 6830	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
16	15	unieqd 4871	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑘))
17	14, 16	eleq12d 2832	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛) ↔ (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘)))
18	17	cbvralvw 3222	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛) ↔ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
19	13, 18	sylibr 233	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
20	19	ad2antll 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
21		vex 3446	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
22	21	elixp 8768	. . . . . . . . 9 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛)))
23	11, 20, 22	sylanbrc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
24	23, 2	eleqtrrdi 2849	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ 𝑋)
25	7	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑋 = ∪ 𝑈)
26	24, 25	eleqtrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ ∪ 𝑈)
27		eluni2 4861	. . . . . 6 ⊢ (𝑓 ∈ ∪ 𝑈 ↔ ∃𝑣 ∈ 𝑈 𝑓 ∈ 𝑣)
28	26, 27	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∃𝑣 ∈ 𝑈 𝑓 ∈ 𝑣)
29		simplrr 776	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ 𝑣)
30	29	adantr 482	. . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . 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 6829	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑓 → (𝑤‘𝑘) = (𝑓‘𝑘))
34	33	eleq1d 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑓 → ((𝑤‘𝑘) ∈ 𝑢 ↔ (𝑓‘𝑘) ∈ 𝑢))
35		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
36	35	mptpreima 6181	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝑋 ∣ (𝑤‘𝑘) ∈ 𝑢}
37	34, 36	elrab2 3641	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ 𝑋 ∧ (𝑓‘𝑘) ∈ 𝑢))
38	37	simprbi 498	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑣 ∈ 𝑈)
42	41	adantr 482	. . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . 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 3424	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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	eleqtrrdi 2849	. . . . . . . . . . . . . 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 4862	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑘) ∈ 𝑢 ∧ 𝑢 ∈ 𝐾) → (𝑓‘𝑘) ∈ ∪ 𝐾)
48	39, 46, 47	syl2anc 585	. . . . . . . . . . . . 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 3150	. . . . . . . . . . . 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 458	. . . . . . . . . . 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 3161	. . . . . . . . . 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 414	. . . . . . . . 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 456	. . . . . . 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 408	. . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑈 ⊆ ran 𝑆)
57	56	sselda 3936	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ ran 𝑆)
58	57	adantrr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) → 𝑣 ∈ ran 𝑆)
59	1	rnmpo 7474	. . . . . . . 8 ⊢ ran 𝑆 = {𝑣 ∣ ∃𝑘 ∈ 𝐴 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)}
60	58, 59	eleqtrdi 2848	. . . . . . 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 2718	. . . . . . 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 217	. . . . . 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 3087	. . . . . 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 3155	. . . 4 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾)
66		eldifn 4079	. . . . . . 7 ⊢ ((𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
67	66	ralimi 3083	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
68	67	ad2antll 727	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
69		ralnex 3073	. . . . 5 ⊢ (∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾 ↔ ¬ ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾)
70	68, 69	sylib 217	. . . 4 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ¬ ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾)
71	65, 70	pm2.65da 815	. . 3 ⊢ (𝜑 → ¬ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
72	71	nexdv 1939	. 2 ⊢ (𝜑 → ¬ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
73	10, 72	pm2.65i 193	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2714  ∀wral 3062  ∃wrex 3071  {crab 3404   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4552  ∪ cuni 4857   ↦ cmpt 5180  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   “ cima 5628   Fn wfn 6479  ⟶wf 6480  ‘cfv 6484   ∈ cmpo 7344  Xcixp 8761  Fincfn 8809  cardccrd 9797  Compccmp 22643  UFLcufl 23157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-int 4900  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-se 5581  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-isom 6493  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-om 7786  df-1st 7904  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-oadd 8376  df-omul 8377  df-er 8574  df-map 8693  df-ixp 8762  df-en 8810  df-dom 8811  df-fin 8813  df-wdom 9427  df-card 9801  df-acn 9804  df-cmp 22644

This theorem is referenced by:  ptcmplem5  23313