Theorem ixpiunwdom 9548

Description: Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8922 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 3459	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
2	1	elixp 8898	. . . . . . . . 9 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
3	2	simprbi 502	. . . . . . . 8 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)
4		ssiun2 5012	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
5	4	sseld 3936	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
6	5	ralimia 3099	. . . . . . . 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 1944	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵
9		nfiu1 4992	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
10	9	nfel2 2943	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵
11		fveq2 6881	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
12	11	eleq1d 2848	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
13	8, 10, 12	cbvralw 3307	. . . . . . 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 486	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) → ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
16	15	ralrimiva 3157	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∀𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
17		eqid 2763	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) = (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦))
18	17	fmpo 8061	. . . 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 8921	. . . . . 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 7443	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴) ∈ V
23	22	ssex 5291	. . . . 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 7746	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (X𝑥 ∈ 𝐴 𝐵 × 𝐴) ∈ V)
27	19, 26	fexd 7225	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) ∈ V)
28	19	ffnd 6706	. . . 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 6798	. . . 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 4960	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
33		nfixp1 8912	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵
34	33	nfel2 2943	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵
35		nfv 1944	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)
36	33, 35	nfrexw 3313	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)
37		simplrr 789	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → 𝑧 ∈ 𝐵)
38		iftrue 4493	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) = 𝑧)
39		csbeq1a 3867	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
40	39	equcoms 2050	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
41	40	eqcomd 2769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑥 → ⦋𝑘 / 𝑥⦌𝐵 = 𝐵)
42	38, 41	eleq12d 2857	. . . . . . . . . . . . . . . . . . . 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 3459	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑔 ∈ V
45	44	elixp 8898	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
46	45	simprbi 502	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
47	46	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
48		nfv 1944	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝑔‘𝑥) ∈ 𝐵
49		nfcsb1v 3877	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵
50	49	nfel2 2943	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑥(𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵
51		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑘 → (𝑔‘𝑥) = (𝑔‘𝑘))
52	51, 39	eleq12d 2857	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑘 → ((𝑔‘𝑥) ∈ 𝐵 ↔ (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵))
53	48, 50, 52	cbvralw 3307	. . . . . . . . . . . . . . . . . . . . . 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 3257	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵)
56		iffalse 4496	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) = (𝑔‘𝑘))
57	56	eleq1d 2848	. . . . . . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵)
61		ixpfn 8897	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑔 Fn 𝐴)
62	61	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑔 Fn 𝐴)
63	62	fndmd 6640	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → dom 𝑔 = 𝐴)
64	44	dmex 7902	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝑔 ∈ V
65	63, 64	eqeltrrdi 2872	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝐴 ∈ V)
66		mptelixpg 8929	. . . . . . . . . . . . . . . . . 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 2925	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝐵
70	69, 49, 39	cbvixp 8908	. . . . . . . . . . . . . . . 16 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑘 ∈ 𝐴 ⦋𝑘 / 𝑥⦌𝐵
71	68, 70	eleqtrrdi 2874	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) ∈ X𝑥 ∈ 𝐴 𝐵)
72		simprl 782	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐴)
73		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) = (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))
74		vex 3459	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
75	38, 73, 74	fvmpt 6989	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥) = 𝑧)
76	75	ad2antrl 740	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥) = 𝑧)
77	76	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥))
78		fveq1 6880	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) → (𝑓‘𝑦) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦))
79	78	eqeq2d 2774	. . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥))
81	80	eqeq2d 2774	. . . . . . . . . . . . . . . 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 3594	. . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦))))
85	34, 36, 84	rexlimd 3272	. . . . . . . . . . . 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 1960	. . . . . . . . . 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 1957	. . . . . . 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 4017	. . . . . . 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 7543	. . . . . 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 3978	. . . . 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 6714	. . . . 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 3954	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 = ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)))
97		foeq3 6790	. . . 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 9529	. 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 595	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 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455  ⦋csb 3853   ⊆ wss 3905  ∅c0 4286  ifcif 4487  ∪ ciun 4956   class class class wbr 5109   ↦ cmpt 5192   × cxp 5659  dom cdm 5661  ran crn 5662   Fn wfn 6531  ⟶wf 6532  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ↑_m cmap 8820  Xcixp 8891   ≼^* cwdom 9522

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-ixp 8892  df-wdom 9523

This theorem is used by:  ptcmplem2  24219