Theorem ptcmplem2 22950

Description: Lemma for ptcmp 22955. (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 4311	. . . . . . 7 ⊢ ∅ ⊆ 𝑈
3		0fin 8849	. . . . . . 7 ⊢ ∅ ∈ Fin
4		elfpw 8978	. . . . . . 7 ⊢ (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ⊆ 𝑈 ∧ ∅ ∈ Fin))
5	2, 3, 4	mpbir2an 711	. . . . . 6 ⊢ ∅ ∈ (𝒫 𝑈 ∩ Fin)
6		unieq 4830	. . . . . . . 8 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∪ ∅)
7		uni0 4849	. . . . . . . 8 ⊢ ∪ ∅ = ∅
8	6, 7	eqtrdi 2794	. . . . . . 7 ⊢ (𝑧 = ∅ → ∪ 𝑧 = ∅)
9	8	rspceeqv 3552	. . . . . 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 2967	. . . 4 ⊢ (¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = ∪ 𝑧 → 𝑋 ≠ ∅)
12	1, 11	syl 17	. . 3 ⊢ (𝜑 → 𝑋 ≠ ∅)
13		n0 4261	. . 3 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ 𝑋)
14	12, 13	sylib 221	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ 𝑋)
15		ptcmp.2	. . . . . . 7 ⊢ 𝑋 = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
16		fveq2 6717	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
17	16	unieqd 4833	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑘))
18	17	cbvixpv 8596	. . . . . . 7 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
19	15, 18	eqtri 2765	. . . . . 6 ⊢ 𝑋 = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
20		ptcmp.5	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (UFL ∩ dom card))
21	20	elin2d 4113	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ dom card)
22	21	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ∈ dom card)
23	19, 22	eqeltrrid 2843	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ dom card)
24		ssrab2 3993	. . . . . 6 ⊢ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ⊆ 𝐴
25	12	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ≠ ∅)
26	19, 25	eqnetrrid 3016	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ≠ ∅)
27		eqid 2737	. . . . . . 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 8617	. . . . . 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 590	. . . . 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 9672	. . . . 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 587	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card)
32		vex 3412	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
33		difexg 5220	. . . . . . . . . . 11 ⊢ (𝑔 ∈ V → (𝑔 ∖ 𝑓) ∈ V)
34	32, 33	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑔 ∖ 𝑓) ∈ V)
35		dmexg 7681	. . . . . . . . . 10 ⊢ ((𝑔 ∖ 𝑓) ∈ V → dom (𝑔 ∖ 𝑓) ∈ V)
36		uniexg 7528	. . . . . . . . . 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 3106	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∀𝑔 ∈ 𝑋 ∪ dom (𝑔 ∖ 𝑓) ∈ V)
39		eqid 2737	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) = (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓))
40	39	fnmpt 6518	. . . . . . . 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 6639	. . . . . . 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 221	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)):𝑋–onto→ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)))
44		fonum 9672	. . . . . 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 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ∈ dom card)
46		ssdif0 4278	. . . . . . . . . . . 12 ⊢ (∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)} ↔ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅)
47		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)})
48		simpr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋)
49	48, 19	eleqtrdi 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
50		vex 3412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑓 ∈ V
51	50	elixp 8585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘)))
52	51	simprbi 500	. . . . . . . . . . . . . . . . . . 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 3130	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (𝑓‘𝑘) ∈ ∪ (𝐹‘𝑘))
55	54	snssd 4722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → {(𝑓‘𝑘)} ⊆ ∪ (𝐹‘𝑘))
56	55	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → {(𝑓‘𝑘)} ⊆ ∪ (𝐹‘𝑘))
57	47, 56	eqssd 3918	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → ∪ (𝐹‘𝑘) = {(𝑓‘𝑘)})
58		fvex 6730	. . . . . . . . . . . . . . 15 ⊢ (𝑓‘𝑘) ∈ V
59	58	ensn1 8694	. . . . . . . . . . . . . 14 ⊢ {(𝑓‘𝑘)} ≈ 1o
60	57, 59	eqbrtrdi 5092	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ ∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)}) → ∪ (𝐹‘𝑘) ≈ 1o)
61	60	ex 416	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (∪ (𝐹‘𝑘) ⊆ {(𝑓‘𝑘)} → ∪ (𝐹‘𝑘) ≈ 1o))
62	46, 61	syl5bir 246	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → ((∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅ → ∪ (𝐹‘𝑘) ≈ 1o))
63	62	con3d 155	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (¬ ∪ (𝐹‘𝑘) ≈ 1o → ¬ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅))
64		neq0 4260	. . . . . . . . . 10 ⊢ (¬ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) = ∅ ↔ ∃𝑥 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}))
65	63, 64	syl6ib 254	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (¬ ∪ (𝐹‘𝑘) ≈ 1o → ∃𝑥 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})))
66		eldifi 4041	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → 𝑥 ∈ ∪ (𝐹‘𝑘))
67		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → 𝑥 ∈ ∪ (𝐹‘𝑘))
68		iftrue 4445	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) = 𝑥)
69	68, 17	eleq12d 2832	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛) ↔ 𝑥 ∈ ∪ (𝐹‘𝑘)))
70	67, 69	syl5ibrcom 250	. . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛))
72	50	elixp 8585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑛 ∈ 𝐴 (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛)))
73	72	simprbi 500	. . . . . . . . . . . . . . . . . . . . 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 3130	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛))
77		iffalse 4448	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑛 = 𝑘 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) = (𝑓‘𝑛))
78	77	eleq1d 2822	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑛 = 𝑘 → (if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛) ↔ (𝑓‘𝑛) ∈ ∪ (𝐹‘𝑛)))
79	76, 78	syl5ibrcom 250	. . . . . . . . . . . . . . . . 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 182	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) ∧ 𝑛 ∈ 𝐴) → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ ∪ (𝐹‘𝑛))
81	80	ralrimiva 3105	. . . . . . . . . . . . . . 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 8616	. . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ ∪ (𝐹‘𝑘)) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ 𝑋)
88	66, 87	sylan2 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∈ 𝑋)
89		vex 3412	. . . . . . . . . . . . . 14 ⊢ 𝑘 ∈ V
90	89	unisn 4841	. . . . . . . . . . . . 13 ⊢ ∪ {𝑘} = 𝑘
91		simplr 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → 𝑘 ∈ 𝐴)
92		eleq1w 2820	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑘 → (𝑚 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
93	91, 92	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑚 = 𝑘 → 𝑚 ∈ 𝐴))
94	93	pm4.71rd 566	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑚 = 𝑘 ↔ (𝑚 ∈ 𝐴 ∧ 𝑚 = 𝑘)))
95		equequ1 2033	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (𝑛 = 𝑘 ↔ 𝑚 = 𝑘))
96		fveq2 6717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
97	95, 96	ifbieq2d 4465	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) = if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)))
98		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))
99		vex 3412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑥 ∈ V
100		fvex 6730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓‘𝑚) ∈ V
101	99, 100	ifex 4489	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ∈ V
102	97, 98, 101	fvmpt 6818	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ 𝐴 → ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) = if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)))
103	102	neeq1d 3000	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ 𝐴 → (((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)))‘𝑚) ≠ (𝑓‘𝑚) ↔ if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚)))
104	103	adantl 485	. . . . . . . . . . . . . . . . . . . 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)))
105		iffalse 4448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) = (𝑓‘𝑚))
106	105	necon1ai 2968	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚) → 𝑚 = 𝑘)
107		eldifsni 4703	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → 𝑥 ≠ (𝑓‘𝑘))
108	107	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → 𝑥 ≠ (𝑓‘𝑘))
109		iftrue 4445	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) = 𝑥)
110		fveq2 6717	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝑓‘𝑚) = (𝑓‘𝑘))
111	109, 110	neeq12d 3002	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑘 → (if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚) ↔ 𝑥 ≠ (𝑓‘𝑘)))
112	108, 111	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → (𝑚 = 𝑘 → if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚)))
113	106, 112	impbid2 229	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) ∧ 𝑚 ∈ 𝐴) → (if(𝑚 = 𝑘, 𝑥, (𝑓‘𝑚)) ≠ (𝑓‘𝑚) ↔ 𝑚 = 𝑘))
114	104, 113	bitrd 282	. . . . . . . . . . . . . . . . . . 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))
115	114	pm5.32da 582	. . . . . . . . . . . . . . . . . 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)))
116	94, 115	bitr4d 285	. . . . . . . . . . . . . . . . 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))))
117	116	abbidv 2807	. . . . . . . . . . . . . . . 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))})
118		df-sn 4542	. . . . . . . . . . . . . . . 16 ⊢ {𝑘} = {𝑚 ∣ 𝑚 = 𝑘}
119		df-rab 3070	. . . . . . . . . . . . . . . 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))}
120	117, 118, 119	3eqtr4g 2803	. . . . . . . . . . . . . . 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)})
121		fvex 6730	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑛) ∈ V
122	99, 121	ifex 4489	. . . . . . . . . . . . . . . . . 18 ⊢ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ V
123	122	rgenw 3073	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ V
124	98	fnmpt 6518	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑛 ∈ 𝐴 if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛)) ∈ V → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) Fn 𝐴)
125	123, 124	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) Fn 𝐴)
126		ixpfn 8584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) → 𝑓 Fn 𝐴)
127	71, 126	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐴)
128	127	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → 𝑓 Fn 𝐴)
129		fndmdif 6862	. . . . . . . . . . . . . . . 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)})
130	125, 128, 129	syl2anc 587	. . . . . . . . . . . . . . 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)})
131	120, 130	eqtr4d 2780	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → {𝑘} = dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
132	131	unieqd 4833	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → ∪ {𝑘} = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
133	90, 132	eqtr3id 2792	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → 𝑘 = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
134		difeq1 4030	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) → (𝑔 ∖ 𝑓) = ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
135	134	dmeqd 5774	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) → dom (𝑔 ∖ 𝑓) = dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
136	135	unieqd 4833	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) → ∪ dom (𝑔 ∖ 𝑓) = ∪ dom ((𝑛 ∈ 𝐴 ↦ if(𝑛 = 𝑘, 𝑥, (𝑓‘𝑛))) ∖ 𝑓))
137	136	rspceeqv 3552	. . . . . . . . . . . 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))
138	88, 133, 137	syl2anc 587	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)})) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓))
139	138	ex 416	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
140	139	exlimdv 1941	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (∃𝑥 𝑥 ∈ (∪ (𝐹‘𝑘) ∖ {(𝑓‘𝑘)}) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
141	65, 140	syld 47	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → (¬ ∪ (𝐹‘𝑘) ≈ 1o → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
142	141	expimpd 457	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ((𝑘 ∈ 𝐴 ∧ ¬ ∪ (𝐹‘𝑘) ≈ 1o) → ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
143	17	breq1d 5063	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (∪ (𝐹‘𝑛) ≈ 1o ↔ ∪ (𝐹‘𝑘) ≈ 1o))
144	143	notbid 321	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (¬ ∪ (𝐹‘𝑛) ≈ 1o ↔ ¬ ∪ (𝐹‘𝑘) ≈ 1o))
145	144	elrab 3602	. . . . . . 7 ⊢ (𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ↔ (𝑘 ∈ 𝐴 ∧ ¬ ∪ (𝐹‘𝑘) ≈ 1o))
146	39	elrnmpt 5825	. . . . . . . 8 ⊢ (𝑘 ∈ V → (𝑘 ∈ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ↔ ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓)))
147	146	elv 3414	. . . . . . 7 ⊢ (𝑘 ∈ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)) ↔ ∃𝑔 ∈ 𝑋 𝑘 = ∪ dom (𝑔 ∖ 𝑓))
148	142, 145, 147	3imtr4g 299	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} → 𝑘 ∈ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓))))
149	148	ssrdv 3907	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ⊆ ran (𝑔 ∈ 𝑋 ↦ ∪ dom (𝑔 ∖ 𝑓)))
150		ssnum 9653	. . . . 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)
151	45, 149, 150	syl2anc 587	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ∈ dom card)
152		xpnum 9567	. . . 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)
153	31, 151, 152	syl2anc 587	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) × {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}) ∈ dom card)
154	82	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐴 ∈ 𝑉)
155		rabexg 5224	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ∈ V)
156	154, 155	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o} ∈ V)
157		fvex 6730	. . . . . . 7 ⊢ (𝐹‘𝑘) ∈ V
158	157	uniex 7529	. . . . . 6 ⊢ ∪ (𝐹‘𝑘) ∈ V
159	158	rgenw 3073	. . . . 5 ⊢ ∀𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ V
160		iunexg 7736	. . . . 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)
161	156, 159, 160	sylancl 589	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ V)
162		resixp 8614	. . . . . 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))
163	24, 49, 162	sylancr 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ↾ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}) ∈ X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘))
164	163	ne0d 4250	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → X𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ≠ ∅)
165		ixpiunwdom 9206	. . . 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}))
166	156, 161, 164, 165	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}))
167		numwdom 9673	. . 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)
168	153, 166, 167	syl2anc 587	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card)
169	14, 168	exlimddv 1943	1 ⊢ (𝜑 → ∪ 𝑘 ∈ {𝑛 ∈ 𝐴 ∣ ¬ ∪ (𝐹‘𝑛) ≈ 1o}∪ (𝐹‘𝑘) ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2110  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3408   ∖ cdif 3863   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  ifcif 4439  𝒫 cpw 4513  {csn 4541  ∪ cuni 4819  ∪ ciun 4904   class class class wbr 5053   ↦ cmpt 5135   × cxp 5549  ^◡ccnv 5550  dom cdm 5551  ran crn 5552   ↾ cres 5553   “ cima 5554   Fn wfn 6375  ⟶wf 6376  –onto→wfo 6378  ‘cfv 6380   ∈ cmpo 7215  1_oc1o 8195  Xcixp 8578   ≈ cen 8623  Fincfn 8626   ≼^* cwdom 9180  cardccrd 9551  Compccmp 22283  UFLcufl 22797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-omul 8207  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-fin 8630  df-wdom 9181  df-card 9555  df-acn 9558

This theorem is referenced by:  ptcmplem3  22951