Theorem ptcmplem2 23973

Description: Lemma for ptcmp 23978. (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)

Assertion

Ref	Expression
ptcmplem2	⊢ (𝜑 → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card)

Distinct variable groups:   𝑘,𝑛,𝑢,𝑤,𝑧,𝐴   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)

Proof of Theorem ptcmplem2

Dummy variables 𝑓 𝑔 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptcmplem2.7	. . . 4 ⊢ (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)
2		0ss 4359	. . . . . . 7 ⊢ ∅ ⊆ 𝑈
3		0fi 8990	. . . . . . 7 ⊢ ∅ ∈ Fin
4		elfpw 9281	. . . . . . 7 ⊢ (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ⊆ 𝑈 ∧ ∅ ∈ Fin))
5	2, 3, 4	mpbir2an 711	. . . . . 6 ⊢ ∅ ∈ (𝒫 𝑈 ∩ Fin)
6		unieq 4878	. . . . . . . 8 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∪ ∅)
7		uni0 4895	. . . . . . . 8 ⊢ ∪ ∅ = ∅
8	6, 7	eqtrdi 2780	. . . . . . 7 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∅)
9	8	rspceeqv 3608	. . . . . 6 ⊢ ((∅ ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)
10	5, 9	mpan 690	. . . . 5 ⊢ (𝑋 = ∅ → ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧)
11	10	necon3bi 2951	. . . 4 ⊢ (¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧 → 𝑋 ≠ ∅)
12	1, 11	syl 17	. . 3 ⊢ (𝜑 → 𝑋 ≠ ∅)
13		n0 4312	. . 3 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ 𝑋)
14	12, 13	sylib 218	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ 𝑋)
15		ptcmp.2	. . . . . . 7 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
16		fveq2 6840	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
17	16	unieqd 4880	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑘))
18	17	cbvixpv 8865	. . . . . . 7 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
19	15, 18	eqtri 2752	. . . . . 6 ⊢ 𝑋 = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
20		ptcmp.5	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
21	20	elin2d 4164	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ dom card)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ∈ dom card)
23	19, 22	eqeltrrid 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ dom card)
24		ssrab2 4039	. . . . . 6 ⊢ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ⊆ 𝐴
25	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ≠ ∅)
26	19, 25	eqnetrrid 3000	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ≠ ∅)
27		eqid 2729	. . . . . . 7 ⊢ (𝑔 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑔 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o})) = (𝑔 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑔 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}))
28	27	resixpfo 8886	. . . . . 6 ⊢ (({𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ⊆ 𝐴 ∧ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ≠ ∅) → (𝑔 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑔 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o})):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)–onto→X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘))
29	24, 26, 28	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑔 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑔 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o})):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)–onto→X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘))
30		fonum 9987	. . . . 5 ⊢ ((X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ dom card ∧ (𝑔 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↦ (𝑔 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o})):X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)–onto→X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘)) → X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card)
31	23, 29, 30	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card)
32		vex 3448	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
33		difexg 5279	. . . . . . . . . . 11 ⊢ (𝑔 ∈ V → (𝑔 ∖ 𝑓) ∈ V)
34	32, 33	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑔 ∖ 𝑓) ∈ V)
35		dmexg 7857	. . . . . . . . . 10 ⊢ ((𝑔 ∖ 𝑓) ∈ V → dom (𝑔 ∖ 𝑓) ∈ V)
36		uniexg 7696	. . . . . . . . . 10 ⊢ (dom (𝑔 ∖ 𝑓) ∈ V → ∪ dom (𝑔 ∖ 𝑓) ∈ V)
37	34, 35, 36	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∪ dom (𝑔 ∖ 𝑓) ∈ V)
38	37	ralrimivw 3129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∀𝑔 ∈ 𝑋 ∪ dom (𝑔 ∖ 𝑓) ∈ V)
39		eqid 2729	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) = (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓))
40	39	fnmpt 6640	. . . . . . . 8 ⊢ (∀𝑔 ∈ 𝑋 ∪ dom (𝑔 ∖ 𝑓) ∈ V → (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) Fn 𝑋)
41	38, 40	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) Fn 𝑋)
42		dffn4 6760	. . . . . . 7 ⊢ ((𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) Fn 𝑋 ↔ (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)):𝑋–onto→ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)))
43	41, 42	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)):𝑋–onto→ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)))
44		fonum 9987	. . . . . 6 ⊢ ((𝑋 ∈ dom card ∧ (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)):𝑋–onto→ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓))) → ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ∈ dom card)
45	22, 43, 44	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ∈ dom card)
46		ssdif0 4325	. . . . . . . . . . . 12 ⊢ (∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)} ↔ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅)
47		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)})
48		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋)
49	48, 19	eleqtrdi 2838	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
50		vex 3448	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑓 ∈ V
51	50	elixp 8854	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘)))
52	51	simprbi 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
53	49, 52	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
54	53	r19.21bi 3227	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
55	54	snssd 4769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → {(𝑓‘𝑘)} ⊆ ∪ (𝐹‘𝑘))
56	55	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → {(𝑓‘𝑘)} ⊆ ∪ (𝐹‘𝑘))
57	47, 56	eqssd 3961	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → ∪ (𝐹‘𝑘) = {(𝑓‘𝑘)})
58		fvex 6853	. . . . . . . . . . . . . . 15 ⊢ (𝑓‘𝑘) ∈ V
59	58	ensn1 8969	. . . . . . . . . . . . . 14 ⊢ {(𝑓‘𝑘)} ≈ 1o
60	57, 59	eqbrtrdi 5141	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → ∪ (𝐹‘𝑘) ≈ 1o)
61	60	ex 412	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)} → ∪ (𝐹‘𝑘) ≈ 1o))
62	46, 61	biimtrrid 243	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → ((∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅ → ∪ (𝐹‘𝑘) ≈ 1o))
63	62	con3d 152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (¬ ∪ (𝐹‘𝑘) ≈ 1o → ¬ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅))
64		neq0 4311	. . . . . . . . . 10 ⊢ (¬ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅ ↔ ∃𝑥 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}))
65	63, 64	imbitrdi 251	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (¬ ∪ (𝐹‘𝑘) ≈ 1o → ∃𝑥 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})))
66		eldifi 4090	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → 𝑥 ∈ ∪ (𝐹‘𝑘))
67		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → 𝑥 ∈ ∪ (𝐹‘𝑘))
68		iftrue 4490	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) = 𝑥)
69	68, 17	eleq12d 2822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛) ↔ 𝑥 ∈ ∪ (𝐹‘𝑘)))
70	67, 69	syl5ibrcom 247	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛)))
71	48, 15	eleqtrdi 2838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
72	50	elixp 8854	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛)))
73	72	simprbi 496	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
74	71, 73	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
75	74	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
76	75	r19.21bi 3227	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
77		iffalse 4493	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) = (𝑓‘𝑛))
78	77	eleq1d 2813	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛) ↔ (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛)))
79	76, 78	syl5ibrcom 247	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → (¬ 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛)))
80	70, 79	pm2.61d 179	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛))
81	80	ralrimiva 3125	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → ∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛))
82		ptcmp.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
83	82	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → 𝐴 ∈ 𝑉)
84		mptelixpg 8885	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ ∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛)))
85	83, 84	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ ∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛)))
86	81, 85	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
87	86, 15	eleqtrrdi 2839	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ 𝑋)
88	66, 87	sylan2 593	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ 𝑋)
89		unisnv 4887	. . . . . . . . . . . . 13 ⊢ ∪ {𝑘} = 𝑘
90		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → 𝑘 ∈ 𝐴)
91		eleq1w 2811	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑘 → (𝑚 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
92	90, 91	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑚 = 𝑘 → 𝑚 ∈ 𝐴))
93	92	pm4.71rd 562	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚 ∈ 𝐴 ∧ 𝑚 = 𝑘)))
94		equequ1 2025	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (𝑛 = 𝑘 ↔ 𝑚 = 𝑘))
95		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
96	94, 95	ifbieq2d 4511	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) = if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)))
97		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))
98		vex 3448	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑥 ∈ V
99		fvex 6853	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓‘𝑚) ∈ V
100	98, 99	ifex 4535	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ∈ V
101	96, 97, 100	fvmpt 6950	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ 𝐴 → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) = if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)))
102	101	neeq1d 2984	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ 𝐴 → (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚)))
103	102	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚)))
104		iffalse 4493	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) = (𝑓‘𝑚))
105	104	necon1ai 2952	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚) → 𝑚 = 𝑘)
106		eldifsni 4750	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → 𝑥 ≠ (𝑓‘𝑘))
107	106	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → 𝑥 ≠ (𝑓‘𝑘))
108		iftrue 4490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) = 𝑥)
109		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝑓‘𝑚) = (𝑓‘𝑘))
110	108, 109	neeq12d 2986	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑘 → (if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚) ↔ 𝑥 ≠ (𝑓‘𝑘)))
111	107, 110	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚)))
112	105, 111	impbid2 226	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → (if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚) ↔ 𝑚 = 𝑘))
113	103, 112	bitrd 279	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚) ↔ 𝑚 = 𝑘))
114	113	pm5.32da 579	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → ((𝑚 ∈ 𝐴 ∧ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚)) ↔ (𝑚 ∈ 𝐴 ∧ 𝑚 = 𝑘)))
115	93, 114	bitr4d 282	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚 ∈ 𝐴 ∧ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚))))
116	115	abbidv 2795	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → {𝑚 ∣ 𝑚 = 𝑘} = {𝑚 ∣ (𝑚 ∈ 𝐴 ∧ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚))})
117		df-sn 4586	. . . . . . . . . . . . . . . 16 ⊢ {𝑘} = {𝑚 ∣ 𝑚 = 𝑘}
118		df-rab 3403	. . . . . . . . . . . . . . . 16 ⊢ {𝑚 ∈ 𝐴 ∣ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚)} = {𝑚 ∣ (𝑚 ∈ 𝐴 ∧ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚))}
119	116, 117, 118	3eqtr4g 2789	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → {𝑘} = {𝑚 ∈ 𝐴 ∣ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚)})
120		fvex 6853	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑛) ∈ V
121	98, 120	ifex 4535	. . . . . . . . . . . . . . . . . 18 ⊢ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ V
122	121	rgenw 3048	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ V
123	97	fnmpt 6640	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ V → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) Fn 𝐴)
124	122, 123	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) Fn 𝐴)
125		ixpfn 8853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) → 𝑓 Fn 𝐴)
126	71, 125	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐴)
127	126	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → 𝑓 Fn 𝐴)
128		fndmdif 6996	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) Fn 𝐴 ∧ 𝑓 Fn 𝐴) → dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓) = {𝑚 ∈ 𝐴 ∣ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚)})
129	124, 127, 128	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓) = {𝑚 ∈ 𝐴 ∣ ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚)})
130	119, 129	eqtr4d 2767	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → {𝑘} = dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
131	130	unieqd 4880	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → ∪ {𝑘} = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
132	89, 131	eqtr3id 2778	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → 𝑘 = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
133		difeq1 4078	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) → (𝑔 ∖ 𝑓) = ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
134	133	dmeqd 5859	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) → dom (𝑔 ∖ 𝑓) = dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
135	134	unieqd 4880	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) → ∪ dom (𝑔 ∖ 𝑓) = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
136	135	rspceeqv 3608	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ 𝑋 ∧ 𝑘 = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓)) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓))
137	88, 132, 136	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓))
138	137	ex 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
139	138	exlimdv 1933	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (∃𝑥 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
140	65, 139	syld 47	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (¬ ∪ (𝐹‘𝑘) ≈ 1o → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
141	140	expimpd 453	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ((𝑘 ∈ 𝐴 ∧ ¬ ∪ (𝐹‘𝑘) ≈ 1o) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
142	17	breq1d 5112	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (∪ (𝐹‘𝑛) ≈ 1o ↔ ∪ (𝐹‘𝑘) ≈ 1o))
143	142	notbid 318	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (¬ ∪ (𝐹‘𝑛) ≈ 1o ↔ ¬ ∪ (𝐹‘𝑘) ≈ 1o))
144	143	elrab 3656	. . . . . . 7 ⊢ (𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ↔ (𝑘 ∈ 𝐴 ∧ ¬ ∪ (𝐹‘𝑘) ≈ 1o))
145	39	elrnmpt 5911	. . . . . . . 8 ⊢ (𝑘 ∈ V → (𝑘 ∈ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ↔ ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
146	145	elv 3449	. . . . . . 7 ⊢ (𝑘 ∈ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ↔ ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓))
147	141, 144, 146	3imtr4g 296	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} → 𝑘 ∈ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓))))
148	147	ssrdv 3949	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ⊆ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)))
149		ssnum 9968	. . . . 5 ⊢ ((ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ∈ dom card ∧ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ⊆ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓))) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ∈ dom card)
150	45, 148, 149	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ∈ dom card)
151		xpnum 9880	. . . 4 ⊢ ((X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card ∧ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ∈ dom card) → (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}) ∈ dom card)
152	31, 150, 151	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}) ∈ dom card)
153	82	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐴 ∈ 𝑉)
154		rabexg 5287	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ∈ V)
155	153, 154	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ∈ V)
156		fvex 6853	. . . . . . 7 ⊢ (𝐹‘𝑘) ∈ V
157	156	uniex 7697	. . . . . 6 ⊢ ∪ (𝐹‘𝑘) ∈ V
158	157	rgenw 3048	. . . . 5 ⊢ ∀𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ V
159		iunexg 7921	. . . . 5 ⊢ (({𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ∈ V ∧ ∀𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ V) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ V)
160	155, 158, 159	sylancl 586	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ V)
161		resixp 8883	. . . . . 6 ⊢ (({𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ⊆ 𝐴 ∧ 𝑓 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)) → (𝑓 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}) ∈ X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘))
162	24, 49, 161	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}) ∈ X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘))
163	162	ne0d 4301	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ≠ ∅)
164		ixpiunwdom 9519	. . . 4 ⊢ (({𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ∈ V ∧ ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ V ∧ X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ≠ ∅) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ≼* (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}))
165	155, 160, 163, 164	syl3anc 1373	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ≼* (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}))
166		numwdom 9988	. . 3 ⊢ (((X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}) ∈ dom card ∧ ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ≼* (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o})) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card)
167	152, 165, 166	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card)
168	14, 167	exlimddv 1935	1 ⊢ (𝜑 → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3402  Vcvv 3444   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  ifcif 4484  𝒫 cpw 4559  {csn 4585  ∪ cuni 4867  ∪ ciun 4951   class class class wbr 5102   ↦ cmpt 5183   × cxp 5629  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634   Fn wfn 6494  ⟶wf 6495  –onto→wfo 6497  ‘cfv 6499   ∈ cmpo 7371  1_oc1o 8404  Xcixp 8847   ≈ cen 8892  Fincfn 8895   ≼^* cwdom 9493  cardccrd 9864  Compccmp 23306  UFLcufl 23820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-oadd 8415  df-omul 8416  df-er 8648  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-fin 8899  df-wdom 9494  df-card 9868  df-acn 9871

This theorem is referenced by:  ptcmplem3  23974