ixpiunwdom - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ixpiunwdom		Structured version Visualization version GIF version

Theorem ixpiunwdom 9584

Description: Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8921 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 3478	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
2	1	elixp 8897	. . . . . . . . 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 5050	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
5	4	sseld 3981	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
6	5	ralimia 3080	. . . . . . . 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 1917	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵
9		nfiu1 5031	. . . . . . . . 9 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
10	9	nfel2 2921	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵
11		fveq2 6891	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
12	11	eleq1d 2818	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
13	8, 10, 12	cbvralw 3303	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
14	7, 13	sylib 217	. . . . . 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 3146	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∀𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
17		eqid 2732	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) = (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦))
18	17	fmpo 8053	. . . 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 217	. . 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 8920	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴))
21	20	3ad2ant2 1134	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴))
22		ovex 7441	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↑_m 𝐴) ∈ V
23	22	ssex 5321	. . . . 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 1136	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → 𝐴 ∈ 𝑉)
26	24, 25	xpexd 7737	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (X𝑥 ∈ 𝐴 𝐵 × 𝐴) ∈ V)
27	19, 26	fexd 7228	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)) ∈ V)
28	19	ffnd 6718	. . . 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 6811	. . . 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 217	. . 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 4346	. . . . . . . . . 10 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵)
32		eliun 5001	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
33		nfixp1 8911	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵
34	33	nfel2 2921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵
35		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)
36	33, 35	nfrexw 3310	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)
37		simplrr 776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → 𝑧 ∈ 𝐵)
38		iftrue 4534	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) = 𝑧)
39		csbeq1a 3907	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
40	39	equcoms 2023	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑥 → 𝐵 = ⦋𝑘 / 𝑥⦌𝐵)
41	40	eqcomd 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑥 → ⦋𝑘 / 𝑥⦌𝐵 = 𝐵)
42	38, 41	eleq12d 2827	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵 ↔ 𝑧 ∈ 𝐵))
43	37, 42	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵))
44		vex 3478	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑔 ∈ V
45	44	elixp 8897	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵))
46	45	simprbi 497	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
47	46	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)
48		nfv 1917	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(𝑔‘𝑥) ∈ 𝐵
49		nfcsb1v 3918	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑥⦋𝑘 / 𝑥⦌𝐵
50	49	nfel2 2921	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑥(𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵
51		fveq2 6891	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑘 → (𝑔‘𝑥) = (𝑔‘𝑘))
52	51, 39	eleq12d 2827	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑘 → ((𝑔‘𝑥) ∈ 𝐵 ↔ (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵))
53	48, 50, 52	cbvralw 3303	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵 ↔ ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵)
54	47, 53	sylib 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑘 ∈ 𝐴 (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵)
55	54	r19.21bi 3248	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵)
56		iffalse 4537	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) = (𝑔‘𝑘))
57	56	eleq1d 2818	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵 ↔ (𝑔‘𝑘) ∈ ⦋𝑘 / 𝑥⦌𝐵))
58	55, 57	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . 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 179	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) ∧ 𝑘 ∈ 𝐴) → if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵)
60	59	ralrimiva 3146	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∀𝑘 ∈ 𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)) ∈ ⦋𝑘 / 𝑥⦌𝐵)
61		ixpfn 8896	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝑔 Fn 𝐴)
62	61	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑔 Fn 𝐴)
63	62	fndmd 6654	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → dom 𝑔 = 𝐴)
64	44	dmex 7901	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝑔 ∈ V
65	63, 64	eqeltrrdi 2842	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝐴 ∈ V)
66		mptelixpg 8928	. . . . . . . . . . . . . . . . . 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 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))
68	60, 67	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) ∈ X𝑘 ∈ 𝐴 ⦋𝑘 / 𝑥⦌𝐵)
69		nfcv 2903	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝐵
70	69, 49, 39	cbvixp 8907	. . . . . . . . . . . . . . . 16 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑘 ∈ 𝐴 ⦋𝑘 / 𝑥⦌𝐵
71	68, 70	eleqtrrdi 2844	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) ∈ X𝑥 ∈ 𝐴 𝐵)
72		simprl 769	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑥 ∈ 𝐴)
73		eqid 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) = (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))
74		vex 3478	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
75	38, 73, 74	fvmpt 6998	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥) = 𝑧)
76	75	ad2antrl 726	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥) = 𝑧)
77	76	eqcomd 2738	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → 𝑧 = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥))
78		fveq1 6890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘))) → (𝑓‘𝑦) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦))
79	78	eqeq2d 2743	. . . . . . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑦) = ((𝑘 ∈ 𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔‘𝑘)))‘𝑥))
81	80	eqeq2d 2743	. . . . . . . . . . . . . . . 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 3624	. . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦))
84	83	exp32 421	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦))))
85	34, 36, 84	rexlimd 3263	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
86	32, 85	biimtrid 241	. . . . . . . . . . 11 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
87	86	exlimiv 1933	. . . . . . . . . 10 ⊢ (∃𝑔 𝑔 ∈ X𝑥 ∈ 𝐴 𝐵 → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
88	31, 87	sylbi 216	. . . . . . . . 9 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ∃𝑓 ∈ X 𝑥 ∈ 𝐴 𝐵∃𝑦 ∈ 𝐴 𝑧 = (𝑓‘𝑦)))
89	88	3ad2ant3 1135	. . . . . . . 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 1930	. . . . . . 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 4058	. . . . . . 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 233	. . . . . 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 7541	. . . . . 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 4033	. . . . 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 6725	. . . . 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 3999	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 = ran (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵, 𝑦 ∈ 𝐴 ↦ (𝑓‘𝑦)))
97		foeq3 6803	. . . 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 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))))
99	30, 98	mpbird 256	. 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 9565	. 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 584	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼^* (X𝑥 ∈ 𝐴 𝐵 × 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2709 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⦋csb 3893 ⊆ wss 3948 ∅c0 4322 ifcif 4528 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 dom cdm 5676 ran crn 5677 Fn wfn 6538 ⟶wf 6539 –onto→wfo 6541 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 ↑_m cmap 8819 Xcixp 8890 ≼^* cwdom 9558

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-map 8821 df-ixp 8891 df-wdom 9559

This theorem is referenced by: ptcmplem2 23556

W3C validator