Theorem ptcmplem4 24313

Description: Lemma for ptcmp 24316. (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 24312	. 2 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
11		simprl 783	. . . . . . . . 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 3099	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
14		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
15		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
16	15	unieqd 4880	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑘))
17	14, 16	eleq12d 2854	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛) ↔ (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘)))
18	17	cbvralvw 3240	. . . . . . . . . . 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 742	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
21		vex 3454	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
22	21	elixp 8918	. . . . . . . . 9 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛)))
23	11, 20, 22	sylanbrc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
24	23, 2	eleqtrrdi 2871	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ 𝑋)
25	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑋 = ∪ 𝑈)
26	24, 25	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ ∪ 𝑈)
27		eluni2 4871	. . . . . 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 790	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑓 ∈ 𝑣)
30	29	adantr 486	. . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . 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 6880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑓 → (𝑤‘𝑘) = (𝑓‘𝑘))
34	33	eleq1d 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑓 → ((𝑤‘𝑘) ∈ 𝑢 ↔ (𝑓‘𝑘) ∈ 𝑢))
35		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘))
36	35	mptpreima 6236	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝑋 ∣ (𝑤‘𝑘) ∈ 𝑢}
37	34, 36	elrab2 3649	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑓 ∈ 𝑋 ∧ (𝑓‘𝑘) ∈ 𝑢))
38	37	simprbi 503	. . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 Fn 𝐴) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) ∧ (𝑘 ∈ 𝐴 ∧ (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑣 ∈ 𝑈)
42	41	adantr 486	. . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . 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 3432	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 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 2871	. . . . . . . . . . . . . 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 4872	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑘) ∈ 𝑢 ∧ 𝑢 ∈ 𝐾) → (𝑓‘𝑘) ∈ ∪ 𝐾)
48	39, 46, 47	syl2anc 596	. . . . . . . . . . . . 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 3164	. . . . . . . . . . . 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 462	. . . . . . . . . . 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 3174	. . . . . . . . . 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 418	. . . . . . . . 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 460	. . . . . . 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 412	. . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → 𝑈 ⊆ ran 𝑆)
57	56	sselda 3931	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ ran 𝑆)
58	57	adantrr 730	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) ∧ (𝑣 ∈ 𝑈 ∧ 𝑓 ∈ 𝑣)) → 𝑣 ∈ ran 𝑆)
59	1	rnmpo 7549	. . . . . . . 8 ⊢ ran 𝑆 = {𝑣 ∣ ∃𝑘 ∈ 𝐴 ∃𝑢 ∈ (𝐹‘𝑘)𝑣 = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)}
60	58, 59	eleqtrdi 2870	. . . . . . 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 2742	. . . . . . 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 3103	. . . . . 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 3169	. . . 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 3099	. . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾) → ∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
68	67	ad2antll 742	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾))) → ∀𝑘 ∈ 𝐴 ¬ (𝑓‘𝑘) ∈ ∪ 𝐾)
69		ralnex 3088	. . . . 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 829	. . 3 ⊢ (𝜑 → ¬ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
72	71	nexdv 1969	. 2 ⊢ (𝜑 → ¬ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ (∪ (𝐹‘𝑘) ∖ ∪ 𝐾)))
73	10, 72	pm2.65i 196	1 ⊢ ¬ 𝜑

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  {crab 3412   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867   ↦ cmpt 5186  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658   Fn wfn 6530  ⟶wf 6531  ‘cfv 6535   ∈ cmpo 7418  Xcixp 8911  Fincfn 8959  cardccrd 9965  Compccmp 23643  UFLcufl 24158

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-oadd 8466  df-omul 8467  df-er 8703  df-map 8835  df-ixp 8912  df-en 8960  df-dom 8961  df-fin 8963  df-wdom 9544  df-card 9969  df-acn 9972  df-cmp 23644

This theorem is used by:  ptcmplem5  24314