kelac1 - Mathbox for Stefan O'Rear

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Theorem kelac1 40408

Description: Kelley's choice, basic form: if a collection of sets can be cast as closed sets in the factors of a topology, and there is a definable element in each topology (which need not be in the closed set - if it were this would be trivial), then compactness (via finite intersection) guarantees that the final product is nonempty. (Contributed by Stefan O'Rear, 22-Feb-2015.)

**Hypotheses**
Ref	Expression
kelac1.z	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)
kelac1.j	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐽 ∈ Top)
kelac1.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐶 ∈ (Clsd‘𝐽))
kelac1.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐵:𝑆–1-1-onto→𝐶)
kelac1.u	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑈 ∈ ∪ 𝐽)
kelac1.k	⊢ (𝜑 → (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Comp)

**Assertion**
Ref	Expression
kelac1	⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)

Distinct variable groups: 𝜑,𝑥 𝑥,𝐼 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑥) 𝑆(𝑥) 𝑈(𝑥) 𝐽(𝑥)

Proof of Theorem kelac1 Dummy variables 𝑓 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		kelac1.c	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐶 ∈ (Clsd‘𝐽))
2		eqid 2758	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cldss 21734	. . . . . . 7 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ ∪ 𝐽)
4	1, 3	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐶 ⊆ ∪ 𝐽)
5	4	ralrimiva 3113	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐶 ⊆ ∪ 𝐽)
6		boxriin 8527	. . . . 5 ⊢ (∀𝑥 ∈ 𝐼 𝐶 ⊆ ∪ 𝐽 → X𝑥 ∈ 𝐼 𝐶 = (X𝑥 ∈ 𝐼 ∪ 𝐽 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝐶 = (X𝑥 ∈ 𝐼 ∪ 𝐽 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
8		kelac1.k	. . . . . . . . 9 ⊢ (𝜑 → (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Comp)
9		cmptop 22100	. . . . . . . . 9 ⊢ ((∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Comp → (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Top)
10		0ntop 21610	. . . . . . . . . . 11 ⊢ ¬ ∅ ∈ Top
11		fvprc 6654	. . . . . . . . . . . 12 ⊢ (¬ (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V → (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) = ∅)
12	11	eleq1d 2836	. . . . . . . . . . 11 ⊢ (¬ (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V → ((∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Top ↔ ∅ ∈ Top))
13	10, 12	mtbiri 330	. . . . . . . . . 10 ⊢ (¬ (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V → ¬ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Top)
14	13	con4i 114	. . . . . . . . 9 ⊢ ((∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Top → (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V)
15	8, 9, 14	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V)
16		kelac1.j	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐽 ∈ Top)
17	16	fmpttd 6875	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶Top)
18		dmfex 7622	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V ∧ (𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶Top) → 𝐼 ∈ V)
19	15, 17, 18	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ V)
20	16	ralrimiva 3113	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐽 ∈ Top)
21		eqid 2758	. . . . . . . 8 ⊢ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) = (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽))
22	21	ptunimpt 22300	. . . . . . 7 ⊢ ((𝐼 ∈ V ∧ ∀𝑥 ∈ 𝐼 𝐽 ∈ Top) → X𝑥 ∈ 𝐼 ∪ 𝐽 = ∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)))
23	19, 20, 22	syl2anc 587	. . . . . 6 ⊢ (𝜑 → X𝑥 ∈ 𝐼 ∪ 𝐽 = ∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)))
24	23	ineq1d 4118	. . . . 5 ⊢ (𝜑 → (X𝑥 ∈ 𝐼 ∪ 𝐽 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)) = (∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
25		eqid 2758	. . . . . 6 ⊢ ∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) = ∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽))
26	2	topcld 21740	. . . . . . . . . 10 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ (Clsd‘𝐽))
27	16, 26	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∪ 𝐽 ∈ (Clsd‘𝐽))
28	1, 27	ifcld 4469	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) ∈ (Clsd‘𝐽))
29	19, 16, 28	ptcldmpt 22319	. . . . . . 7 ⊢ (𝜑 → X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) ∈ (Clsd‘(∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽))))
30	29	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) ∈ (Clsd‘(∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽))))
31		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) → 𝑧 ∈ Fin)
32		kelac1.b	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐵:𝑆–1-1-onto→𝐶)
33		f1ofo 6613	. . . . . . . . . . . . . . 15 ⊢ (𝐵:𝑆–1-1-onto→𝐶 → 𝐵:𝑆–onto→𝐶)
34		foima 6585	. . . . . . . . . . . . . . 15 ⊢ (𝐵:𝑆–onto→𝐶 → (𝐵 “ 𝑆) = 𝐶)
35	32, 33, 34	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐵 “ 𝑆) = 𝐶)
36	35	eqcomd 2764	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐶 = (𝐵 “ 𝑆))
37		kelac1.z	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)
38		f1ofn 6607	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵:𝑆–1-1-onto→𝐶 → 𝐵 Fn 𝑆)
39	32, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐵 Fn 𝑆)
40		ssid 3916	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 ⊆ 𝑆
41		fnimaeq0 6468	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 Fn 𝑆 ∧ 𝑆 ⊆ 𝑆) → ((𝐵 “ 𝑆) = ∅ ↔ 𝑆 = ∅))
42	39, 40, 41	sylancl 589	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐵 “ 𝑆) = ∅ ↔ 𝑆 = ∅))
43	42	necon3bid 2995	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐵 “ 𝑆) ≠ ∅ ↔ 𝑆 ≠ ∅))
44	37, 43	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐵 “ 𝑆) ≠ ∅)
45	36, 44	eqnetrd 3018	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐶 ≠ ∅)
46		n0 4247	. . . . . . . . . . . 12 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐶)
47	45, 46	sylib 221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∃𝑤 𝑤 ∈ 𝐶)
48		rexv 3436	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ V 𝑤 ∈ 𝐶 ↔ ∃𝑤 𝑤 ∈ 𝐶)
49	47, 48	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∃𝑤 ∈ V 𝑤 ∈ 𝐶)
50	49	ralrimiva 3113	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ∃𝑤 ∈ V 𝑤 ∈ 𝐶)
51		ssralv 3960	. . . . . . . . . 10 ⊢ (𝑧 ⊆ 𝐼 → (∀𝑥 ∈ 𝐼 ∃𝑤 ∈ V 𝑤 ∈ 𝐶 → ∀𝑥 ∈ 𝑧 ∃𝑤 ∈ V 𝑤 ∈ 𝐶))
52	51	adantr 484	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin) → (∀𝑥 ∈ 𝐼 ∃𝑤 ∈ V 𝑤 ∈ 𝐶 → ∀𝑥 ∈ 𝑧 ∃𝑤 ∈ V 𝑤 ∈ 𝐶))
53	50, 52	mpan9 510	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) → ∀𝑥 ∈ 𝑧 ∃𝑤 ∈ V 𝑤 ∈ 𝐶)
54		eleq1 2839	. . . . . . . . 9 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤 ∈ 𝐶 ↔ (𝑓‘𝑥) ∈ 𝐶))
55	54	ac6sfi 8800	. . . . . . . 8 ⊢ ((𝑧 ∈ Fin ∧ ∀𝑥 ∈ 𝑧 ∃𝑤 ∈ V 𝑤 ∈ 𝐶) → ∃𝑓(𝑓:𝑧⟶V ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶))
56	31, 53, 55	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) → ∃𝑓(𝑓:𝑧⟶V ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶))
57	23	eqcomd 2764	. . . . . . . . . . 11 ⊢ (𝜑 → ∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) = X𝑥 ∈ 𝐼 ∪ 𝐽)
58	57	ineq1d 4118	. . . . . . . . . 10 ⊢ (𝜑 → (∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∩ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)) = (X𝑥 ∈ 𝐼 ∪ 𝐽 ∩ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
59	58	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → (∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∩ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)) = (X𝑥 ∈ 𝐼 ∪ 𝐽 ∩ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
60		iftrue 4429	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑧 → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) = (𝑓‘𝑥))
61	60	ad2antrl 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ (𝑥 ∈ 𝑧 ∧ (𝑓‘𝑥) ∈ 𝐶)) → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) = (𝑓‘𝑥))
62		simpll 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ 𝑥 ∈ 𝑧) → 𝜑)
63		simprl 770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) → 𝑧 ⊆ 𝐼)
64	63	sselda 3894	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝐼)
65	62, 64, 4	syl2anc 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ 𝑥 ∈ 𝑧) → 𝐶 ⊆ ∪ 𝐽)
66	65	sseld 3893	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ 𝑥 ∈ 𝑧) → ((𝑓‘𝑥) ∈ 𝐶 → (𝑓‘𝑥) ∈ ∪ 𝐽))
67	66	impr 458	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ (𝑥 ∈ 𝑧 ∧ (𝑓‘𝑥) ∈ 𝐶)) → (𝑓‘𝑥) ∈ ∪ 𝐽)
68	61, 67	eqeltrd 2852	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ (𝑥 ∈ 𝑧 ∧ (𝑓‘𝑥) ∈ 𝐶)) → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽)
69	68	expr 460	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ 𝑥 ∈ 𝑧) → ((𝑓‘𝑥) ∈ 𝐶 → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽))
70	69	ralimdva 3108	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) → (∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶 → ∀𝑥 ∈ 𝑧 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽))
71	70	imp 410	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → ∀𝑥 ∈ 𝑧 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽)
72		eldifn 4035	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝐼 ∖ 𝑧) → ¬ 𝑥 ∈ 𝑧)
73	72	iffalsed 4434	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐼 ∖ 𝑧) → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) = 𝑈)
74	73	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) = 𝑈)
75		eldifi 4034	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐼 ∖ 𝑧) → 𝑥 ∈ 𝐼)
76		kelac1.u	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑈 ∈ ∪ 𝐽)
77	75, 76	sylan2 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → 𝑈 ∈ ∪ 𝐽)
78	74, 77	eqeltrd 2852	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽)
79	78	ralrimiva 3113	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ (𝐼 ∖ 𝑧)if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽)
80	79	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → ∀𝑥 ∈ (𝐼 ∖ 𝑧)if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽)
81		ralun 4099	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝑧 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽 ∧ ∀𝑥 ∈ (𝐼 ∖ 𝑧)if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽) → ∀𝑥 ∈ (𝑧 ∪ (𝐼 ∖ 𝑧))if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽)
82	71, 80, 81	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → ∀𝑥 ∈ (𝑧 ∪ (𝐼 ∖ 𝑧))if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽)
83		undif 4381	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ⊆ 𝐼 ↔ (𝑧 ∪ (𝐼 ∖ 𝑧)) = 𝐼)
84	83	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ 𝐼 → (𝑧 ∪ (𝐼 ∖ 𝑧)) = 𝐼)
85	84	ad2antrl 727	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) → (𝑧 ∪ (𝐼 ∖ 𝑧)) = 𝐼)
86	85	raleqdv 3329	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) → (∀𝑥 ∈ (𝑧 ∪ (𝐼 ∖ 𝑧))if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽 ↔ ∀𝑥 ∈ 𝐼 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽))
87	86	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → (∀𝑥 ∈ (𝑧 ∪ (𝐼 ∖ 𝑧))if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽 ↔ ∀𝑥 ∈ 𝐼 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽))
88	82, 87	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → ∀𝑥 ∈ 𝐼 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽)
89	19	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → 𝐼 ∈ V)
90		mptelixpg 8522	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ V → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ X𝑥 ∈ 𝐼 ∪ 𝐽 ↔ ∀𝑥 ∈ 𝐼 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽))
91	89, 90	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ X𝑥 ∈ 𝐼 ∪ 𝐽 ↔ ∀𝑥 ∈ 𝐼 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ ∪ 𝐽))
92	88, 91	mpbird 260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ X𝑥 ∈ 𝐼 ∪ 𝐽)
93		eleq2 2840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶 = if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) → ((𝑓‘𝑥) ∈ 𝐶 ↔ (𝑓‘𝑥) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
94		eleq2 2840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∪ 𝐽 = if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) → ((𝑓‘𝑥) ∈ ∪ 𝐽 ↔ (𝑓‘𝑥) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
95		simplrr 777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ (𝑥 ∈ 𝑧 ∧ (𝑓‘𝑥) ∈ 𝐶)) ∧ 𝑥 = 𝑦) → (𝑓‘𝑥) ∈ 𝐶)
96	67	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ (𝑥 ∈ 𝑧 ∧ (𝑓‘𝑥) ∈ 𝐶)) ∧ ¬ 𝑥 = 𝑦) → (𝑓‘𝑥) ∈ ∪ 𝐽)
97	93, 94, 95, 96	ifbothda 4461	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ (𝑥 ∈ 𝑧 ∧ (𝑓‘𝑥) ∈ 𝐶)) → (𝑓‘𝑥) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
98	61, 97	eqeltrd 2852	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ (𝑥 ∈ 𝑧 ∧ (𝑓‘𝑥) ∈ 𝐶)) → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
99	98	expr 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ 𝑥 ∈ 𝑧) → ((𝑓‘𝑥) ∈ 𝐶 → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
100	99	ralimdva 3108	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) → (∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶 → ∀𝑥 ∈ 𝑧 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
101	100	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → ∀𝑥 ∈ 𝑧 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
102	101	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) ∧ 𝑦 ∈ 𝑧) → ∀𝑥 ∈ 𝑧 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
103	77	adantlr 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → 𝑈 ∈ ∪ 𝐽)
104	73	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) = 𝑈)
105		incom 4108	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐼 ∖ 𝑧) ∩ 𝑧) = (𝑧 ∩ (𝐼 ∖ 𝑧))
106		disjdif 4371	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∩ (𝐼 ∖ 𝑧)) = ∅
107	105, 106	eqtri 2781	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐼 ∖ 𝑧) ∩ 𝑧) = ∅
108	107	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → ((𝐼 ∖ 𝑧) ∩ 𝑧) = ∅)
109		simpr 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → 𝑥 ∈ (𝐼 ∖ 𝑧))
110		simplr 768	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → 𝑦 ∈ 𝑧)
111		disjne 4354	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐼 ∖ 𝑧) ∩ 𝑧) = ∅ ∧ 𝑥 ∈ (𝐼 ∖ 𝑧) ∧ 𝑦 ∈ 𝑧) → 𝑥 ≠ 𝑦)
112	108, 109, 110, 111	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → 𝑥 ≠ 𝑦)
113	112	neneqd 2956	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → ¬ 𝑥 = 𝑦)
114	113	iffalsed 4434	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) = ∪ 𝐽)
115	103, 104, 114	3eltr4d 2867	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑧) ∧ 𝑥 ∈ (𝐼 ∖ 𝑧)) → if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
116	115	ralrimiva 3113	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑧) → ∀𝑥 ∈ (𝐼 ∖ 𝑧)if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
117	116	adantlr 714	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ 𝑦 ∈ 𝑧) → ∀𝑥 ∈ (𝐼 ∖ 𝑧)if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
118	117	adantlr 714	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) ∧ 𝑦 ∈ 𝑧) → ∀𝑥 ∈ (𝐼 ∖ 𝑧)if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
119		ralun 4099	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑥 ∈ 𝑧 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) ∧ ∀𝑥 ∈ (𝐼 ∖ 𝑧)if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)) → ∀𝑥 ∈ (𝑧 ∪ (𝐼 ∖ 𝑧))if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
120	102, 118, 119	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) ∧ 𝑦 ∈ 𝑧) → ∀𝑥 ∈ (𝑧 ∪ (𝐼 ∖ 𝑧))if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
121	85	raleqdv 3329	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) → (∀𝑥 ∈ (𝑧 ∪ (𝐼 ∖ 𝑧))if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) ↔ ∀𝑥 ∈ 𝐼 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
122	121	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) ∧ 𝑦 ∈ 𝑧) → (∀𝑥 ∈ (𝑧 ∪ (𝐼 ∖ 𝑧))if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) ↔ ∀𝑥 ∈ 𝐼 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
123	120, 122	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) ∧ 𝑦 ∈ 𝑧) → ∀𝑥 ∈ 𝐼 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
124	19	ad3antrrr 729	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) ∧ 𝑦 ∈ 𝑧) → 𝐼 ∈ V)
125		mptelixpg 8522	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ V → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) ↔ ∀𝑥 ∈ 𝐼 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
126	124, 125	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) ∧ 𝑦 ∈ 𝑧) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) ↔ ∀𝑥 ∈ 𝐼 if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈) ∈ if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
127	123, 126	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) ∧ 𝑦 ∈ 𝑧) → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
128	127	ralrimiva 3113	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → ∀𝑦 ∈ 𝑧 (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
129		mptexg 6980	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ V → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ V)
130	19, 129	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ V)
131	130	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ V)
132		eliin 4891	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ V → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) ↔ ∀𝑦 ∈ 𝑧 (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
133	131, 132	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽) ↔ ∀𝑦 ∈ 𝑧 (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
134	128, 133	mpbird 260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽))
135	92, 134	elind 4101	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑧, (𝑓‘𝑥), 𝑈)) ∈ (X𝑥 ∈ 𝐼 ∪ 𝐽 ∩ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)))
136	135	ne0d 4236	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → (X𝑥 ∈ 𝐼 ∪ 𝐽 ∩ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)) ≠ ∅)
137	59, 136	eqnetrd 3018	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶) → (∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∩ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)) ≠ ∅)
138	137	adantrl 715	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) ∧ (𝑓:𝑧⟶V ∧ ∀𝑥 ∈ 𝑧 (𝑓‘𝑥) ∈ 𝐶)) → (∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∩ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)) ≠ ∅)
139	56, 138	exlimddv 1936	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐼 ∧ 𝑧 ∈ Fin)) → (∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∩ ∩ 𝑦 ∈ 𝑧 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)) ≠ ∅)
140	25, 8, 30, 139	cmpfiiin 40039	. . . . 5 ⊢ (𝜑 → (∪ (∏_t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)) ≠ ∅)
141	24, 140	eqnetrd 3018	. . . 4 ⊢ (𝜑 → (X𝑥 ∈ 𝐼 ∪ 𝐽 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐶, ∪ 𝐽)) ≠ ∅)
142	7, 141	eqnetrd 3018	. . 3 ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝐶 ≠ ∅)
143		n0 4247	. . 3 ⊢ (X𝑥 ∈ 𝐼 𝐶 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ X𝑥 ∈ 𝐼 𝐶)
144	142, 143	sylib 221	. 2 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ X𝑥 ∈ 𝐼 𝐶)
145		elixp2 8488	. . . . . 6 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 𝐶 ↔ (𝑦 ∈ V ∧ 𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ 𝐶))
146	145	simp3bi 1144	. . . . 5 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 𝐶 → ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ 𝐶)
147		f1ocnv 6618	. . . . . . . 8 ⊢ (𝐵:𝑆–1-1-onto→𝐶 → ^◡𝐵:𝐶–1-1-onto→𝑆)
148		f1of 6606	. . . . . . . 8 ⊢ (^◡𝐵:𝐶–1-1-onto→𝑆 → ^◡𝐵:𝐶⟶𝑆)
149		ffvelrn 6845	. . . . . . . . 9 ⊢ ((^◡𝐵:𝐶⟶𝑆 ∧ (𝑦‘𝑥) ∈ 𝐶) → (^◡𝐵‘(𝑦‘𝑥)) ∈ 𝑆)
150	149	ex 416	. . . . . . . 8 ⊢ (^◡𝐵:𝐶⟶𝑆 → ((𝑦‘𝑥) ∈ 𝐶 → (^◡𝐵‘(𝑦‘𝑥)) ∈ 𝑆))
151	32, 147, 148, 150	4syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝑦‘𝑥) ∈ 𝐶 → (^◡𝐵‘(𝑦‘𝑥)) ∈ 𝑆))
152	151	ralimdva 3108	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ 𝐶 → ∀𝑥 ∈ 𝐼 (^◡𝐵‘(𝑦‘𝑥)) ∈ 𝑆))
153	152	imp 410	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ 𝐶) → ∀𝑥 ∈ 𝐼 (^◡𝐵‘(𝑦‘𝑥)) ∈ 𝑆)
154	146, 153	sylan2 595	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ X𝑥 ∈ 𝐼 𝐶) → ∀𝑥 ∈ 𝐼 (^◡𝐵‘(𝑦‘𝑥)) ∈ 𝑆)
155		mptelixpg 8522	. . . . . 6 ⊢ (𝐼 ∈ V → ((𝑥 ∈ 𝐼 ↦ (^◡𝐵‘(𝑦‘𝑥))) ∈ X𝑥 ∈ 𝐼 𝑆 ↔ ∀𝑥 ∈ 𝐼 (^◡𝐵‘(𝑦‘𝑥)) ∈ 𝑆))
156	19, 155	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ (^◡𝐵‘(𝑦‘𝑥))) ∈ X𝑥 ∈ 𝐼 𝑆 ↔ ∀𝑥 ∈ 𝐼 (^◡𝐵‘(𝑦‘𝑥)) ∈ 𝑆))
157	156	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ X𝑥 ∈ 𝐼 𝐶) → ((𝑥 ∈ 𝐼 ↦ (^◡𝐵‘(𝑦‘𝑥))) ∈ X𝑥 ∈ 𝐼 𝑆 ↔ ∀𝑥 ∈ 𝐼 (^◡𝐵‘(𝑦‘𝑥)) ∈ 𝑆))
158	154, 157	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ X𝑥 ∈ 𝐼 𝐶) → (𝑥 ∈ 𝐼 ↦ (^◡𝐵‘(𝑦‘𝑥))) ∈ X𝑥 ∈ 𝐼 𝑆)
159	158	ne0d 4236	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ X𝑥 ∈ 𝐼 𝐶) → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)
160	144, 159	exlimddv 1936	1 ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2951 ∀wral 3070 ∃wrex 3071 Vcvv 3409 ∖ cdif 3857 ∪ cun 3858 ∩ cin 3859 ⊆ wss 3860 ∅c0 4227 ifcif 4423 ∪ cuni 4801 ∩ ciin 4887 ↦ cmpt 5115 ^◡ccnv 5526 “ cima 5530 Fn wfn 6334 ⟶wf 6335 –onto→wfo 6337 –1-1-onto→wf1o 6338 ‘cfv 6339 Xcixp 8484 Fincfn 8532 ∏_tcpt 16775 Topctop 21598 Clsdccld 21721 Compccmp 22091

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-om 7585 df-1o 8117 df-er 8304 df-ixp 8485 df-en 8533 df-fin 8536 df-fi 8913 df-topgen 16780 df-pt 16781 df-top 21599 df-bases 21651 df-cld 21724 df-cmp 22092

This theorem is referenced by: kelac2 40410

W3C validator