Theorem ptcmplem4 24183

Description: Lemma for ptcmp 24186. (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 24182	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
11		simprl 782	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 Fn 𝐴)
12		eldifi 4093	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
13	12	ralimi 3108	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
14		fveq2 6884	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
15		fveq2 6884	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
16	15	unieqd 4889	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑘))
17	14, 16	eleq12d 2863	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛) ↔ (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘)))
18	17	cbvralvw 3249	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛) ↔ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
19	13, 18	sylibr 237	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
20	19	ad2antll 741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
21		vex 3467	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
22	21	elixp 8904	. . . . . . . . 9 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛)))
23	11, 20, 22	sylanbrc 594	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
24	23, 2	eleqtrrdi 2880	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ 𝑋)
25	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑋 = ∪ 𝑈)
26	24, 25	eleqtrd 2871	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ ∪ 𝑈)
27		eluni2 4880	. . . . . 6 ⊢ (𝑓 ∈ ∪ 𝑈 ↔ ∃𝑣 ∈ 𝑈 𝑓 ∈ 𝑣)
28	26, 27	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∃𝑣 ∈ 𝑈 𝑓 ∈ 𝑣)
29		simplrr 789	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ 𝑣)
30	29	adantr 485	. . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . 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 2871	. . . . . . . . . . . . . . 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 6883	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑓 → (𝑤‘𝑘) = (𝑓‘𝑘))
34	33	eleq1d 2854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑓 → ((𝑤‘𝑘) ∈ 𝑢 ↔ (𝑓‘𝑘) ∈ 𝑢))
35		eqid 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
36	35	mptpreima 6242	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝑋 ∣ (𝑤‘𝑘) ∈ 𝑢}
37	34, 36	elrab2 3663	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ 𝑋 ∧ (𝑓‘𝑘) ∈ 𝑢))
38	37	simprbi 502	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) → (𝑓‘𝑘) ∈ 𝑢)
39	32, 38	syl 18	. . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑣 ∈ 𝑈)
42	41	adantr 485	. . . . . . . . . . . . . . . . 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 2870	. . . . . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . 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 2880	. . . . . . . . . . . . . 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 4881	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑘) ∈ 𝑢 ∧ 𝑢 ∈ 𝐾) → (𝑓‘𝑘) ∈ ∪ 𝐾)
48	39, 46, 47	syl2anc 595	. . . . . . . . . . . . 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 3173	. . . . . . . . . . . 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 461	. . . . . . . . . . 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 3183	. . . . . . . . . 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 417	. . . . . . . . 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 87	. . . . . . . 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 459	. . . . . . 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 411	. . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑈 ⊆ ran 𝑆)
57	56	sselda 3945	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ ran 𝑆)
58	57	adantrr 729	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) → 𝑣 ∈ ran 𝑆)
59	1	rnmpo 7546	. . . . . . . 8 ⊢ ran 𝑆 = {𝑣 ∣ ∃𝑘 ∈ 𝐴 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)}
60	58, 59	eleqtrdi 2879	. . . . . . 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 2751	. . . . . . 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 221	. . . . . 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 3112	. . . . . 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 66	. . . . 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 3178	. . . 4 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾)
66		eldifn 4094	. . . . . . 7 ⊢ ((𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
67	66	ralimi 3108	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
68	67	ad2antll 741	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
69		ralnex 3097	. . . . 5 ⊢ (∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾 ↔ ¬ ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾)
70	68, 69	sylib 221	. . . 4 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ¬ ∃𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ 𝐾)
71	65, 70	pm2.65da 828	. . 3 ⊢ (𝜑 → ¬ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
72	71	nexdv 1963	. 2 ⊢ (𝜑 → ¬ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
73	10, 72	pm2.65i 196	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095  {crab 3423   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4876   ↦ cmpt 5196  ^◡ccnv 5663  dom cdm 5664  ran crn 5665   “ cima 5667   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539   ∈ cmpo 7415  Xcixp 8897  Fincfn 8945  cardccrd 9923  Compccmp 23514  UFLcufl 24028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-omul 8460  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-fin 8949  df-wdom 9529  df-card 9927  df-acn 9930  df-cmp 23515

This theorem is referenced by:  ptcmplem5  24184