Theorem ixpiunwdom 8901

Description: Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8340 this shows that ∪ 𝑥 ∈ 𝐴𝐵 and X𝑥 ∈ 𝐴𝐵 have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.)

Assertion

Ref	Expression
ixpiunwdom	⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼* (X𝑥 ∈ 𝐴 𝐵 × 𝐴))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ixpiunwdom

Dummy variables 𝑓 𝑔 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3440	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
2	1	elixp 8317	. . . . . . . . 9 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
3	2	simprbi 497	. . . . . . . 8 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)
4		ssiun2 4870	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
5	4	sseld 3888	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
6	5	ralimia 3125	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
7	3, 6	syl 17	. . . . . . 7 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
8		nfv 1892	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵
9		nfiu1 4856	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
10	9	nfel2 2965	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵
11		fveq2 6538	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
12	11	eleq1d 2867	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
13	8, 10, 12	cbvral 3399	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
14	7, 13	sylib 219	. . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
15	14	adantl 482	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
16	15	ralrimiva 3149	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∀𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
17		eqid 2795	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) = (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦))
18	17	fmpo 7622	. . . 4 ⊢ (∀𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)⟶∪ 𝑥 ∈ 𝐴 𝐵)
19	16, 18	sylib 219	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)⟶∪ 𝑥 ∈ 𝐴 𝐵)
20		ixpssmap2g 8339	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴))
21	20	3ad2ant2 1127	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴))
22		ovex 7048	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴) ∈ V
23	22	ssex 5116	. . . . 5 ⊢ (X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴) → X𝑥 ∈ 𝐴 𝐵 ∈ V)
24	21, 23	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → X𝑥 ∈ 𝐴 𝐵 ∈ V)
25		simp1 1129	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → 𝐴 ∈ 𝑉)
26	24, 25	xpexd 7331	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (X𝑥 ∈ 𝐴 𝐵 × 𝐴) ∈ V)
27		simp2 1130	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊)
28		fex2 7494	. . 3 ⊢ (((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)⟶∪ 𝑥 ∈ 𝐴 𝐵 ∧ (X𝑥 ∈ 𝐴 𝐵 × 𝐴) ∈ V ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) ∈ V)
29	19, 26, 27, 28	syl3anc 1364	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) ∈ V)
30	19	ffnd 6383	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) Fn (X𝑥 ∈ 𝐴 𝐵 × 𝐴))
31		dffn4 6464	. . . 4 ⊢ ((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) Fn (X𝑥 ∈ 𝐴 𝐵 × 𝐴) ↔ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)–onto→ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)))
32	30, 31	sylib 219	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)–onto→ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)))
33		n0 4230	. . . . . . . . . 10 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)
34		eliun 4829	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
35		nfixp1 8330	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵
36	35	nfel2 2965	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵
37		nfv 1892	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)
38	35, 37	nfrex 3271	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)
39		simplrr 774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → 𝑧 ∈ 𝐵)
40		iftrue 4387	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) = 𝑧)
41		csbeq1a 3824	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
42	41	equcoms 2004	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
43	42	eqcomd 2801	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑥 → ⦋𝑘 / 𝑥⦌𝐵 = 𝐵)
44	40, 43	eleq12d 2877	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵 ↔ 𝑧 ∈ 𝐵))
45	39, 44	syl5ibrcom 248	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵))
46		vex 3440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑔 ∈ V
47	46	elixp 8317	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
48	47	simprbi 497	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
49	48	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
50		nfv 1892	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝑔‘𝑥) ∈ 𝐵
51		nfcsb1v 3833	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵
52	51	nfel2 2965	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑥(𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵
53		fveq2 6538	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑘 → (𝑔‘𝑥) = (𝑔‘𝑘))
54	53, 41	eleq12d 2877	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑘 → ((𝑔‘𝑥) ∈ 𝐵 ↔ (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵))
55	50, 52, 54	cbvral 3399	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵 ↔ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵)
56	49, 55	sylib 219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵)
57	56	r19.21bi 3175	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵)
58		iffalse 4390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) = (𝑔‘𝑘))
59	58	eleq1d 2867	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵 ↔ (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵))
60	57, 59	syl5ibrcom 248	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → (¬ 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵))
61	45, 60	pm2.61d 180	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵)
62	61	ralrimiva 3149	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵)
63		ixpfn 8316	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑔 Fn 𝐴)
64	63	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑔 Fn 𝐴)
65		fndm 6325	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 Fn 𝐴 → dom 𝑔 = 𝐴)
66	64, 65	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → dom 𝑔 = 𝐴)
67	46	dmex 7472	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝑔 ∈ V
68	66, 67	syl6eqelr 2892	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝐴 ∈ V)
69		mptelixpg 8347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ V → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) ∈ X𝑘 ∈ 𝐴 ⦋𝑘 / 𝑥⦌𝐵 ↔ ∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵))
70	68, 69	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) ∈ X𝑘 ∈ 𝐴 ⦋𝑘 / 𝑥⦌𝐵 ↔ ∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵))
71	62, 70	mpbird 258	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) ∈ X𝑘 ∈ 𝐴 ⦋𝑘 / 𝑥⦌𝐵)
72		nfcv 2949	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝐵
73	72, 51, 41	cbvixp 8327	. . . . . . . . . . . . . . . 16 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑘 ∈ 𝐴 ⦋𝑘 / 𝑥⦌𝐵
74	71, 73	syl6eleqr 2894	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) ∈ X𝑥 ∈ 𝐴 𝐵)
75		simprl 767	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐴)
76		eqid 2795	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) = (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))
77		vex 3440	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
78	40, 76, 77	fvmpt 6635	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥) = 𝑧)
79	78	ad2antrl 724	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥) = 𝑧)
80	79	eqcomd 2801	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥))
81		fveq1 6537	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) → (𝑓‘𝑦) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦))
82	81	eqeq2d 2805	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) → (𝑧 = (𝑓‘𝑦) ↔ 𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦)))
83		fveq2 6538	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥))
84	83	eqeq2d 2805	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦) ↔ 𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥)))
85	82, 84	rspc2ev 3574	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥)) → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦))
86	74, 75, 80, 85	syl3anc 1364	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦))
87	86	exp32 421	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦))))
88	36, 38, 87	rexlimd 3278	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
89	34, 88	syl5bi 243	. . . . . . . . . . 11 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
90	89	exlimiv 1908	. . . . . . . . . 10 ⊢ (∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
91	33, 90	sylbi 218	. . . . . . . . 9 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
92	91	3ad2ant3 1128	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
93	92	alrimiv 1905	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∀𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
94		ssab 3962	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ {𝑧 ∣ ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)} ↔ ∀𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
95	93, 94	sylibr 235	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ {𝑧 ∣ ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)})
96	17	rnmpo 7140	. . . . . 6 ⊢ ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) = {𝑧 ∣ ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)}
97	95, 96	syl6sseqr 3939	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)))
98	19	frnd 6389	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
99	97, 98	eqssd 3906	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 = ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)))
100		foeq3 6456	. . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 = ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) → ((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)–onto→∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)–onto→ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦))))
101	99, 100	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)–onto→∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)–onto→ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦))))
102	32, 101	mpbird 258	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)–onto→∪ 𝑥 ∈ 𝐴 𝐵)
103		fowdom 8881	. 2 ⊢ (((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) ∈ V ∧ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)):(X𝑥 ∈ 𝐴 𝐵 × 𝐴)–onto→∪ 𝑥 ∈ 𝐴 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼* (X𝑥 ∈ 𝐴 𝐵 × 𝐴))
104	29, 102, 103	syl2anc 584	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼* (X𝑥 ∈ 𝐴 𝐵 × 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080  ∀wal 1520   = wceq 1522  ∃wex 1761   ∈ wcel 2081  {cab 2775   ≠ wne 2984  ∀wral 3105  ∃wrex 3106  Vcvv 3437  ⦋csb 3811   ⊆ wss 3859  ∅c0 4211  ifcif 4381  ∪ ciun 4825   class class class wbr 4962   ↦ cmpt 5041   × cxp 5441  dom cdm 5443  ran crn 5444   Fn wfn 6220  ⟶wf 6221  –onto→wfo 6223  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018   ↑_𝑚 cmap 8256  Xcixp 8310   ≼^* cwdom 8867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-map 8258  df-ixp 8311  df-wdom 8869

This theorem is referenced by:  ptcmplem2  22345