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Theorem ixpiunwdom 8440
Description: Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 7882 this shows that 𝑥𝐴𝐵 and X𝑥𝐴𝐵 have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.)
Assertion
Ref Expression
ixpiunwdom ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵* (X𝑥𝐴 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.
StepHypRef Expression
1 vex 3189 . . . . . . . . . 10 𝑓 ∈ V
21elixp 7859 . . . . . . . . 9 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simprbi 480 . . . . . . . 8 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
4 ssiun2 4529 . . . . . . . . . 10 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
54sseld 3582 . . . . . . . . 9 (𝑥𝐴 → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ 𝑥𝐴 𝐵))
65ralimia 2945 . . . . . . . 8 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵)
73, 6syl 17 . . . . . . 7 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵)
8 nfv 1840 . . . . . . . 8 𝑦(𝑓𝑥) ∈ 𝑥𝐴 𝐵
9 nfiu1 4516 . . . . . . . . 9 𝑥 𝑥𝐴 𝐵
109nfel2 2777 . . . . . . . 8 𝑥(𝑓𝑦) ∈ 𝑥𝐴 𝐵
11 fveq2 6148 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
1211eleq1d 2683 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝑥𝐴 𝐵 ↔ (𝑓𝑦) ∈ 𝑥𝐴 𝐵))
138, 10, 12cbvral 3155 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
147, 13sylib 208 . . . . . 6 (𝑓X𝑥𝐴 𝐵 → ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
1514adantl 482 . . . . 5 (((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) ∧ 𝑓X𝑥𝐴 𝐵) → ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
1615ralrimiva 2960 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ∀𝑓X 𝑥𝐴 𝐵𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
17 eqid 2621 . . . . 5 (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) = (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦))
1817fmpt2 7182 . . . 4 (∀𝑓X 𝑥𝐴 𝐵𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵 ↔ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)⟶ 𝑥𝐴 𝐵)
1916, 18sylib 208 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)⟶ 𝑥𝐴 𝐵)
20 ixpssmap2g 7881 . . . . . 6 ( 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
21203ad2ant2 1081 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
22 ovex 6632 . . . . . 6 ( 𝑥𝐴 𝐵𝑚 𝐴) ∈ V
2322ssex 4762 . . . . 5 (X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴) → X𝑥𝐴 𝐵 ∈ V)
2421, 23syl 17 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → X𝑥𝐴 𝐵 ∈ V)
25 simp1 1059 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝐴𝑉)
26 xpexg 6913 . . . 4 ((X𝑥𝐴 𝐵 ∈ V ∧ 𝐴𝑉) → (X𝑥𝐴 𝐵 × 𝐴) ∈ V)
2724, 25, 26syl2anc 692 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (X𝑥𝐴 𝐵 × 𝐴) ∈ V)
28 simp2 1060 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵𝑊)
29 fex2 7068 . . 3 (((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)⟶ 𝑥𝐴 𝐵 ∧ (X𝑥𝐴 𝐵 × 𝐴) ∈ V ∧ 𝑥𝐴 𝐵𝑊) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ∈ V)
3019, 27, 28, 29syl3anc 1323 . 2 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ∈ V)
31 ffn 6002 . . . . 5 ((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)⟶ 𝑥𝐴 𝐵 → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) Fn (X𝑥𝐴 𝐵 × 𝐴))
3219, 31syl 17 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) Fn (X𝑥𝐴 𝐵 × 𝐴))
33 dffn4 6078 . . . 4 ((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) Fn (X𝑥𝐴 𝐵 × 𝐴) ↔ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto→ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
3432, 33sylib 208 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto→ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
35 n0 3907 . . . . . . . . . 10 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑔 𝑔X𝑥𝐴 𝐵)
36 eliun 4490 . . . . . . . . . . . 12 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
37 nfixp1 7872 . . . . . . . . . . . . . 14 𝑥X𝑥𝐴 𝐵
3837nfel2 2777 . . . . . . . . . . . . 13 𝑥 𝑔X𝑥𝐴 𝐵
39 nfv 1840 . . . . . . . . . . . . . 14 𝑥𝑦𝐴 𝑧 = (𝑓𝑦)
4037, 39nfrex 3001 . . . . . . . . . . . . 13 𝑥𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)
41 simplrr 800 . . . . . . . . . . . . . . . . . . . 20 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → 𝑧𝐵)
42 iftrue 4064 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) = 𝑧)
43 csbeq1a 3523 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘𝐵 = 𝑘 / 𝑥𝐵)
4443equcoms 1944 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑥𝐵 = 𝑘 / 𝑥𝐵)
4544eqcomd 2627 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑥𝑘 / 𝑥𝐵 = 𝐵)
4642, 45eleq12d 2692 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵𝑧𝐵))
4741, 46syl5ibrcom 237 . . . . . . . . . . . . . . . . . . 19 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
48 vex 3189 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑔 ∈ V
4948elixp 7859 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔X𝑥𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
5049simprbi 480 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
5150adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
52 nfv 1840 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑔𝑥) ∈ 𝐵
53 nfcsb1v 3530 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥𝑘 / 𝑥𝐵
5453nfel2 2777 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥(𝑔𝑘) ∈ 𝑘 / 𝑥𝐵
55 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑘 → (𝑔𝑥) = (𝑔𝑘))
5655, 43eleq12d 2692 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘 → ((𝑔𝑥) ∈ 𝐵 ↔ (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵))
5752, 54, 56cbvral 3155 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵 ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
5851, 57sylib 208 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
5958r19.21bi 2927 . . . . . . . . . . . . . . . . . . . 20 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
60 iffalse 4067 . . . . . . . . . . . . . . . . . . . . 21 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) = (𝑔𝑘))
6160eleq1d 2683 . . . . . . . . . . . . . . . . . . . 20 𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵 ↔ (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵))
6259, 61syl5ibrcom 237 . . . . . . . . . . . . . . . . . . 19 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (¬ 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
6347, 62pm2.61d 170 . . . . . . . . . . . . . . . . . 18 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵)
6463ralrimiva 2960 . . . . . . . . . . . . . . . . 17 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵)
65 ixpfn 7858 . . . . . . . . . . . . . . . . . . . . 21 (𝑔X𝑥𝐴 𝐵𝑔 Fn 𝐴)
6665adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑔 Fn 𝐴)
67 fndm 5948 . . . . . . . . . . . . . . . . . . . 20 (𝑔 Fn 𝐴 → dom 𝑔 = 𝐴)
6866, 67syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → dom 𝑔 = 𝐴)
6948dmex 7046 . . . . . . . . . . . . . . . . . . 19 dom 𝑔 ∈ V
7068, 69syl6eqelr 2707 . . . . . . . . . . . . . . . . . 18 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝐴 ∈ V)
71 mptelixpg 7889 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ V → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑘𝐴 𝑘 / 𝑥𝐵 ↔ ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
7270, 71syl 17 . . . . . . . . . . . . . . . . 17 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑘𝐴 𝑘 / 𝑥𝐵 ↔ ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
7364, 72mpbird 247 . . . . . . . . . . . . . . . 16 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑘𝐴 𝑘 / 𝑥𝐵)
74 nfcv 2761 . . . . . . . . . . . . . . . . 17 𝑘𝐵
7574, 53, 43cbvixp 7869 . . . . . . . . . . . . . . . 16 X𝑥𝐴 𝐵 = X𝑘𝐴 𝑘 / 𝑥𝐵
7673, 75syl6eleqr 2709 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑥𝐴 𝐵)
77 simprl 793 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑥𝐴)
78 eqid 2621 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))
79 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
8042, 78, 79fvmpt 6239 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥) = 𝑧)
8180ad2antrl 763 . . . . . . . . . . . . . . . 16 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥) = 𝑧)
8281eqcomd 2627 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥))
83 fveq1 6147 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) → (𝑓𝑦) = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦))
8483eqeq2d 2631 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) → (𝑧 = (𝑓𝑦) ↔ 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦)))
85 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦) = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥))
8685eqeq2d 2631 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦) ↔ 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥)))
8784, 86rspc2ev 3308 . . . . . . . . . . . . . . 15 (((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑥𝐴 𝐵𝑥𝐴𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥)) → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))
8876, 77, 82, 87syl3anc 1323 . . . . . . . . . . . . . 14 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))
8988exp32 630 . . . . . . . . . . . . 13 (𝑔X𝑥𝐴 𝐵 → (𝑥𝐴 → (𝑧𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))))
9038, 40, 89rexlimd 3019 . . . . . . . . . . . 12 (𝑔X𝑥𝐴 𝐵 → (∃𝑥𝐴 𝑧𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9136, 90syl5bi 232 . . . . . . . . . . 11 (𝑔X𝑥𝐴 𝐵 → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9291exlimiv 1855 . . . . . . . . . 10 (∃𝑔 𝑔X𝑥𝐴 𝐵 → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9335, 92sylbi 207 . . . . . . . . 9 (X𝑥𝐴 𝐵 ≠ ∅ → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
94933ad2ant3 1082 . . . . . . . 8 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9594alrimiv 1852 . . . . . . 7 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ∀𝑧(𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
96 ssab 3651 . . . . . . 7 ( 𝑥𝐴 𝐵 ⊆ {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)} ↔ ∀𝑧(𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9795, 96sylibr 224 . . . . . 6 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 ⊆ {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)})
9817rnmpt2 6723 . . . . . 6 ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) = {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)}
9997, 98syl6sseqr 3631 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 ⊆ ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
100 frn 6010 . . . . . 6 ((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)⟶ 𝑥𝐴 𝐵 → ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ⊆ 𝑥𝐴 𝐵)
10119, 100syl 17 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ⊆ 𝑥𝐴 𝐵)
10299, 101eqssd 3600 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 = ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
103 foeq3 6070 . . . 4 ( 𝑥𝐴 𝐵 = ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) → ((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto 𝑥𝐴 𝐵 ↔ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto→ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦))))
104102, 103syl 17 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto 𝑥𝐴 𝐵 ↔ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto→ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦))))
10534, 104mpbird 247 . 2 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto 𝑥𝐴 𝐵)
106 fowdom 8420 . 2 (((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ∈ V ∧ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵* (X𝑥𝐴 𝐵 × 𝐴))
10730, 105, 106syl2anc 692 1 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵* (X𝑥𝐴 𝐵 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  Vcvv 3186  csb 3514  wss 3555  c0 3891  ifcif 4058   ciun 4485   class class class wbr 4613  cmpt 4673   × cxp 5072  dom cdm 5074  ran crn 5075   Fn wfn 5842  wf 5843  ontowfo 5845  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802  Xcixp 7852  * cwdom 8406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-ixp 7853  df-wdom 8408
This theorem is referenced by:  ptcmplem2  21767
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