Theorem ixpiunwdom 9559

Description: Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8932 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 3461	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
2	1	elixp 8908	. . . . . . . . 9 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
3	2	simprbi 503	. . . . . . . 8 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)
4		ssiun2 5014	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
5	4	sseld 3937	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
6	5	ralimia 3101	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
7	3, 6	syl 18	. . . . . . 7 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
8		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵
9		nfiu1 4994	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
10	9	nfel2 2945	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵
11		fveq2 6885	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
12	11	eleq1d 2850	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
13	8, 10, 12	cbvralw 3309	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
14	7, 13	sylib 221	. . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
15	14	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
16	15	ralrimiva 3159	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∀𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
17		eqid 2765	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) = (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦))
18	17	fmpo 8071	. . . 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 221	. . 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 8931	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴))
21	20	3ad2ant2 1152	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴))
22		ovex 7452	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) ∈ V
23	22	ssex 5293	. . . . 5 ⊢ (X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) → X𝑥 ∈ 𝐴 𝐵 ∈ V)
24	21, 23	syl 18	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → X𝑥 ∈ 𝐴 𝐵 ∈ V)
25		simp1 1154	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → 𝐴 ∈ 𝑉)
26	24, 25	xpexd 7756	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (X𝑥 ∈ 𝐴 𝐵 × 𝐴) ∈ V)
27	19, 26	fexd 7232	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) ∈ V)
28	19	ffnd 6710	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) Fn (X𝑥 ∈ 𝐴 𝐵 × 𝐴))
29		dffn4 6802	. . . 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)))
30	28, 29	sylib 221	. . 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)))
31		n0 4307	. . . . . . . . . 10 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)
32		eliun 4962	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
33		nfixp1 8922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵
34	33	nfel2 2945	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵
35		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)
36	33, 35	nfrexw 3315	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)
37		simplrr 790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → 𝑧 ∈ 𝐵)
38		iftrue 4495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) = 𝑧)
39		csbeq1a 3868	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
40	39	equcoms 2053	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
41	40	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑥 → ⦋𝑘 / 𝑥⦌𝐵 = 𝐵)
42	38, 41	eleq12d 2859	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵 ↔ 𝑧 ∈ 𝐵))
43	37, 42	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵))
44		vex 3461	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑔 ∈ V
45	44	elixp 8908	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
46	45	simprbi 503	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
47	46	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
48		nfv 1947	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝑔‘𝑥) ∈ 𝐵
49		nfcsb1v 3878	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵
50	49	nfel2 2945	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑥(𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵
51		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑘 → (𝑔‘𝑥) = (𝑔‘𝑘))
52	51, 39	eleq12d 2859	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑘 → ((𝑔‘𝑥) ∈ 𝐵 ↔ (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵))
53	48, 50, 52	cbvralw 3309	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵 ↔ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵)
54	47, 53	sylib 221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵)
55	54	r19.21bi 3259	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵)
56		iffalse 4498	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) = (𝑔‘𝑘))
57	56	eleq1d 2850	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵 ↔ (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵))
58	55, 57	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → (¬ 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵))
59	43, 58	pm2.61d 181	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵)
60	59	ralrimiva 3159	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵)
61		ixpfn 8907	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑔 Fn 𝐴)
62	61	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑔 Fn 𝐴)
63	62	fndmd 6644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → dom 𝑔 = 𝐴)
64	44	dmex 7912	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝑔 ∈ V
65	63, 64	eqeltrrdi 2874	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝐴 ∈ V)
66		mptelixpg 8939	. . . . . . . . . . . . . . . . . 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))
67	65, 66	syl 18	. . . . . . . . . . . . . . . . 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))
68	60, 67	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) ∈ X𝑘 ∈ 𝐴 ⦋𝑘 / 𝑥⦌𝐵)
69		nfcv 2927	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝐵
70	69, 49, 39	cbvixp 8918	. . . . . . . . . . . . . . . 16 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑘 ∈ 𝐴 ⦋𝑘 / 𝑥⦌𝐵
71	68, 70	eleqtrrdi 2876	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) ∈ X𝑥 ∈ 𝐴 𝐵)
72		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐴)
73		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) = (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))
74		vex 3461	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
75	38, 73, 74	fvmpt 6993	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥) = 𝑧)
76	75	ad2antrl 741	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥) = 𝑧)
77	76	eqcomd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥))
78		fveq1 6884	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) → (𝑓‘𝑦) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦))
79	78	eqeq2d 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) → (𝑧 = (𝑓‘𝑦) ↔ 𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦)))
80		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥))
81	80	eqeq2d 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦) ↔ 𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥)))
82	79, 81	rspc2ev 3596	. . . . . . . . . . . . . . 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))
83	71, 72, 77, 82	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦))
84	83	exp32 426	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦))))
85	34, 36, 84	rexlimd 3274	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
86	32, 85	biimtrid 245	. . . . . . . . . . 11 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
87	86	exlimiv 1963	. . . . . . . . . 10 ⊢ (∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
88	31, 87	sylbi 220	. . . . . . . . 9 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
89	88	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
90	89	alrimiv 1960	. . . . . . 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)))
91		ssab 4018	. . . . . . 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)))
92	90, 91	sylibr 237	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ {𝑧 ∣ ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)})
93	17	rnmpo 7552	. . . . . 6 ⊢ ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) = {𝑧 ∣ ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)}
94	92, 93	sseqtrrdi 3979	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)))
95	19	frnd 6718	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
96	94, 95	eqssd 3955	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 = ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)))
97		foeq3 6794	. . . 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))))
98	96, 97	syl 18	. . 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))))
99	30, 98	mpbird 260	. 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)
100		fowdom 9540	. 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))
101	27, 99, 100	syl2anc 596	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼* (X𝑥 ∈ 𝐴 𝐵 × 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  Vcvv 3457  ⦋csb 3854   ⊆ wss 3906  ∅c0 4286  ifcif 4489  ∪ ciun 4958   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661  dom cdm 5663  ran crn 5664   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421   ↑_m cmap 8830  Xcixp 8901   ≼^* cwdom 9533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-ixp 8902  df-wdom 9534

This theorem is used by:  ptcmplem2  24265