Theorem esumiun 34257

Description: Sum over a nonnecessarily disjoint indexed union. The inequality is strict in the case where the sets B(x) overlap. (Contributed by Thierry Arnoux, 21-Sep-2019.)

Hypotheses

Ref	Expression
esumiun.0	⊢ (𝜑 → 𝐴 ∈ 𝑉)
esumiun.1	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
esumiun.2	⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))

Assertion

Ref	Expression
esumiun	⊢ (𝜑 → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶)

Distinct variable groups:   𝐴,𝑗,𝑘   𝐵,𝑘   𝐶,𝑗   𝑗,𝑊,𝑘   𝜑,𝑗,𝑘

Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑘)   𝑉(𝑗,𝑘)

Proof of Theorem esumiun

Dummy variables 𝑓 𝑙 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		esumiun.0	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		esumiun.1	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
3	1, 2	aciunf1 32754	. . 3 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
4		f1f1orn 6786	. . . . . 6 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
5	4	anim1i 616	. . . . 5 ⊢ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
6		f1f 6731	. . . . . . 7 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵⟶∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
7	6	frnd 6671	. . . . . 6 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
8	7	adantr 480	. . . . 5 ⊢ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
9	5, 8	jca 511	. . . 4 ⊢ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
10	9	eximi 1837	. . 3 ⊢ (∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → ∃𝑓((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
11	3, 10	syl 17	. 2 ⊢ (𝜑 → ∃𝑓((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
12		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑧(𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
13		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑧𝐶
14		nfcsb1v 3862	. . . . . 6 ⊢ Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌𝐶
15		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑧∪ 𝑗 ∈ 𝐴 𝐵
16		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑧ran 𝑓
17		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑧◡𝑓
18		csbeq1a 3852	. . . . . 6 ⊢ (𝑘 = (2nd ‘𝑧) → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
19	2	ralrimiva 3130	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊)
20		iunexg 7910	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
21	1, 19, 20	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ∪ 𝑗 ∈ 𝐴 𝐵 ∈ V)
23		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
24		f1ocnv 6787	. . . . . . . 8 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑗 ∈ 𝐴 𝐵)
25	23, 24	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑗 ∈ 𝐴 𝐵)
26	25	adantrlr 724	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ◡𝑓:ran 𝑓–1-1-onto→∪ 𝑗 ∈ 𝐴 𝐵)
27		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
28		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗𝑓
29		nfiu1 4970	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 𝐵
30	28	nfrn 5902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗ran 𝑓
31	28, 29, 30	nff1o 6773	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓
32		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(2nd ‘(𝑓‘𝑙)) = 𝑙
33	29, 32	nfralw 3285	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙
34	31, 33	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
35		nfcv 2899	. . . . . . . . . . 11 ⊢ Ⅎ𝑗ran 𝑓
36		nfiu1 4970	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
37	35, 36	nfss 3915	. . . . . . . . . 10 ⊢ Ⅎ𝑗ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
38	34, 37	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
39	27, 38	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
40		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑧 ∈ ran 𝑓
41	39, 40	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑗((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓)
42		simpr 484	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (𝑓‘𝑘) = 𝑧)
43	42	fveq2d 6839	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘(𝑓‘𝑘)) = (2nd ‘𝑧))
44		simplr 769	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
45		simp-4r 784	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
46	45	simpld 494	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙))
47	46	simprd 495	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
48	47	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙)
49		2fveq3 6840	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → (2nd ‘(𝑓‘𝑙)) = (2nd ‘(𝑓‘𝑘)))
50		id 22	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘)
51	49, 50	eqeq12d 2753	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → ((2nd ‘(𝑓‘𝑙)) = 𝑙 ↔ (2nd ‘(𝑓‘𝑘)) = 𝑘))
52	51	rspcva 3563	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) → (2nd ‘(𝑓‘𝑘)) = 𝑘)
53	44, 48, 52	syl2anc 585	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘(𝑓‘𝑘)) = 𝑘)
54	43, 53	eqtr3d 2774	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (2nd ‘𝑧) = 𝑘)
55	46	simpld 494	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
56	55	ad2antrr 727	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓)
57		f1ocnvfv1 7225	. . . . . . . . . 10 ⊢ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
58	56, 44, 57	syl2anc 585	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘(𝑓‘𝑘)) = 𝑘)
59	42	fveq2d 6839	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘(𝑓‘𝑘)) = (◡𝑓‘𝑧))
60	54, 58, 59	3eqtr2rd 2779	. . . . . . . 8 ⊢ (((((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ (𝑓‘𝑘) = 𝑧) → (◡𝑓‘𝑧) = (2nd ‘𝑧))
61		f1ofn 6776	. . . . . . . . . 10 ⊢ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 → 𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵)
62	55, 61	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵)
63		simpllr 776	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝑧 ∈ ran 𝑓)
64		fvelrnb 6895	. . . . . . . . . 10 ⊢ (𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵 → (𝑧 ∈ ran 𝑓 ↔ ∃𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧))
65	64	biimpa 476	. . . . . . . . 9 ⊢ ((𝑓 Fn ∪ 𝑗 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ran 𝑓) → ∃𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧)
66	62, 63, 65	syl2anc 585	. . . . . . . 8 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(𝑓‘𝑘) = 𝑧)
67	60, 66	r19.29a 3146	. . . . . . 7 ⊢ (((((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (◡𝑓‘𝑧) = (2nd ‘𝑧))
68		simprr 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
69	68	sselda 3922	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
70		eliun 4938	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
71	69, 70	sylib 218	. . . . . . 7 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
72	41, 67, 71	r19.29af 3247	. . . . . 6 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ran 𝑓) → (◡𝑓‘𝑧) = (2nd ‘𝑧))
73		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑘
74	73, 29	nfel 2914	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵
75	27, 74	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵)
76		esumiun.2	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
77	76	adantllr 720	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
78		eliun 4938	. . . . . . . . . 10 ⊢ (𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 ↔ ∃𝑗 ∈ 𝐴 𝑘 ∈ 𝐵)
79	78	biimpi 216	. . . . . . . . 9 ⊢ (𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵 → ∃𝑗 ∈ 𝐴 𝑘 ∈ 𝐵)
80	79	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → ∃𝑗 ∈ 𝐴 𝑘 ∈ 𝐵)
81	75, 77, 80	r19.29af 3247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
82	81	adantlr 716	. . . . . 6 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵) → 𝐶 ∈ (0[,]+∞))
83	12, 13, 14, 15, 16, 17, 18, 22, 26, 72, 82	esumf1o 34213	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 = Σ*𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
84	83	eqcomd 2743	. . . 4 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶)
85		vsnex 5373	. . . . . . . . . 10 ⊢ {𝑗} ∈ V
86	85	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → {𝑗} ∈ V)
87	86, 2	xpexd 7699	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ({𝑗} × 𝐵) ∈ V)
88	87	ralrimiva 3130	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
89		iunexg 7910	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
90	1, 88, 89	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
91	90	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∈ V)
92		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑧
93	92, 36	nfel 2914	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
94	27, 93	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
95		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑗(2nd ‘𝑧)
96		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐶
97	95, 96	nfcsbw 3864	. . . . . . . 8 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌𝐶
98		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑗(0[,]+∞)
99	97, 98	nfel 2914	. . . . . . 7 ⊢ Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞)
100		simprr 773	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → (2nd ‘𝑧) ∈ 𝐵)
101		simplll 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝜑)
102		simplr 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → 𝑗 ∈ 𝐴)
103	76	ralrimiva 3130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
104	101, 102, 103	syl2anc 585	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
105		rspcsbela 4379	. . . . . . . 8 ⊢ (((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
106	100, 104, 105	syl2anc 585	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
107		xp1st 7968	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗})
108		elsni 4585	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑧) ∈ {𝑗} → (1st ‘𝑧) = 𝑗)
109	107, 108	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗)
110		xp2nd 7969	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
111	109, 110	jca 511	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
112	111	reximi 3076	. . . . . . . . 9 ⊢ (∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
113	70, 112	sylbi 217	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
114	113	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ∃𝑗 ∈ 𝐴 ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) ∈ 𝐵))
115	94, 99, 106, 114	r19.29af2 3246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
116	115	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ∈ (0[,]+∞))
117		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
118	117	adantrlr 724	. . . . 5 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
119	12, 91, 116, 118	esummono 34217	. . . 4 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑧 ∈ ran 𝑓⦋(2nd ‘𝑧) / 𝑘⦌𝐶 ≤ Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
120	84, 119	eqbrtrrd 5110	. . 3 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
121		vex 3434	. . . . . . . . 9 ⊢ 𝑗 ∈ V
122		vex 3434	. . . . . . . . 9 ⊢ 𝑘 ∈ V
123	121, 122	op2ndd 7947	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → (2nd ‘𝑧) = 𝑘)
124	123	eqcomd 2743	. . . . . . 7 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝑘 = (2nd ‘𝑧))
125	124, 18	syl 17	. . . . . 6 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐶 = ⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
126	125	eqcomd 2743	. . . . 5 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → ⦋(2nd ‘𝑧) / 𝑘⦌𝐶 = 𝐶)
127	76	anasss 466	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
128	14, 126, 1, 2, 127	esum2d 34256	. . . 4 ⊢ (𝜑 → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
129	128	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)⦋(2nd ‘𝑧) / 𝑘⦌𝐶)
130	120, 129	breqtrrd 5114	. 2 ⊢ ((𝜑 ∧ ((𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1-onto→ran 𝑓 ∧ ∀𝑙 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑙)) = 𝑙) ∧ ran 𝑓 ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))) → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶)
131	11, 130	exlimddv 1937	1 ⊢ (𝜑 → Σ*𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵𝐶 ≤ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430  ⦋csb 3838   ⊆ wss 3890  {csn 4568  ⟨cop 4574  ∪ ciun 4934   class class class wbr 5086   × cxp 5623  ^◡ccnv 5624  ran crn 5626   Fn wfn 6488  –1-1→wf1 6490  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7361  1^st c1st 7934  2^nd c2nd 7935  0cc0 11032  +∞cpnf 11170   ≤ cle 11174  [,]cicc 13295  Σ^*cesum 34190

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-reg 9501  ax-inf2 9556  ax-ac2 10379  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110  ax-addf 11111  ax-mulf 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-fi 9318  df-sup 9349  df-inf 9350  df-oi 9419  df-r1 9682  df-rank 9683  df-card 9857  df-ac 10032  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-q 12893  df-rp 12937  df-xneg 13057  df-xadd 13058  df-xmul 13059  df-ioo 13296  df-ioc 13297  df-ico 13298  df-icc 13299  df-fz 13456  df-fzo 13603  df-fl 13745  df-mod 13823  df-seq 13958  df-exp 14018  df-fac 14230  df-bc 14259  df-hash 14287  df-shft 15023  df-cj 15055  df-re 15056  df-im 15057  df-sqrt 15191  df-abs 15192  df-limsup 15427  df-clim 15444  df-rlim 15445  df-sum 15643  df-ef 16026  df-sin 16028  df-cos 16029  df-pi 16031  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-starv 17229  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-unif 17237  df-hom 17238  df-cco 17239  df-rest 17379  df-topn 17380  df-0g 17398  df-gsum 17399  df-topgen 17400  df-pt 17401  df-prds 17404  df-ordt 17459  df-xrs 17460  df-qtop 17465  df-imas 17466  df-xps 17468  df-mre 17542  df-mrc 17543  df-acs 17545  df-ps 18526  df-tsr 18527  df-plusf 18601  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19038  df-subg 19093  df-cntz 19286  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-cring 20211  df-subrng 20517  df-subrg 20541  df-abv 20780  df-lmod 20851  df-scaf 20852  df-sra 21163  df-rgmod 21164  df-psmet 21339  df-xmet 21340  df-met 21341  df-bl 21342  df-mopn 21343  df-fbas 21344  df-fg 21345  df-cnfld 21348  df-top 22872  df-topon 22889  df-topsp 22911  df-bases 22924  df-cld 22997  df-ntr 22998  df-cls 22999  df-nei 23076  df-lp 23114  df-perf 23115  df-cn 23205  df-cnp 23206  df-haus 23293  df-tx 23540  df-hmeo 23733  df-fil 23824  df-fm 23916  df-flim 23917  df-flf 23918  df-tmd 24050  df-tgp 24051  df-tsms 24105  df-trg 24138  df-xms 24298  df-ms 24299  df-tms 24300  df-nm 24560  df-ngp 24561  df-nrg 24563  df-nlm 24564  df-ii 24857  df-cncf 24858  df-limc 25846  df-dv 25847  df-log 26536  df-esum 34191

This theorem is referenced by:  omssubadd  34463