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Theorem ixpiunwdom 9526
Description: Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8866 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 3449 . . . . . . . . . 10 𝑓 ∈ V
21elixp 8842 . . . . . . . . 9 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simprbi 497 . . . . . . . 8 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
4 ssiun2 5007 . . . . . . . . . 10 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
54sseld 3943 . . . . . . . . 9 (𝑥𝐴 → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ 𝑥𝐴 𝐵))
65ralimia 3083 . . . . . . . 8 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵)
73, 6syl 17 . . . . . . 7 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵)
8 nfv 1917 . . . . . . . 8 𝑦(𝑓𝑥) ∈ 𝑥𝐴 𝐵
9 nfiu1 4988 . . . . . . . . 9 𝑥 𝑥𝐴 𝐵
109nfel2 2925 . . . . . . . 8 𝑥(𝑓𝑦) ∈ 𝑥𝐴 𝐵
11 fveq2 6842 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
1211eleq1d 2822 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝑥𝐴 𝐵 ↔ (𝑓𝑦) ∈ 𝑥𝐴 𝐵))
138, 10, 12cbvralw 3289 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
147, 13sylib 217 . . . . . 6 (𝑓X𝑥𝐴 𝐵 → ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
1514adantl 482 . . . . 5 (((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) ∧ 𝑓X𝑥𝐴 𝐵) → ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
1615ralrimiva 3143 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ∀𝑓X 𝑥𝐴 𝐵𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
17 eqid 2736 . . . . 5 (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) = (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦))
1817fmpo 8000 . . . 4 (∀𝑓X 𝑥𝐴 𝐵𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵 ↔ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)⟶ 𝑥𝐴 𝐵)
1916, 18sylib 217 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)⟶ 𝑥𝐴 𝐵)
20 ixpssmap2g 8865 . . . . . 6 ( 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
21203ad2ant2 1134 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
22 ovex 7390 . . . . . 6 ( 𝑥𝐴 𝐵m 𝐴) ∈ V
2322ssex 5278 . . . . 5 (X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴) → X𝑥𝐴 𝐵 ∈ V)
2421, 23syl 17 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → X𝑥𝐴 𝐵 ∈ V)
25 simp1 1136 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝐴𝑉)
2624, 25xpexd 7685 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (X𝑥𝐴 𝐵 × 𝐴) ∈ V)
2719, 26fexd 7177 . 2 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ∈ V)
2819ffnd 6669 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) Fn (X𝑥𝐴 𝐵 × 𝐴))
29 dffn4 6762 . . . 4 ((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) Fn (X𝑥𝐴 𝐵 × 𝐴) ↔ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto→ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
3028, 29sylib 217 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto→ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
31 n0 4306 . . . . . . . . . 10 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑔 𝑔X𝑥𝐴 𝐵)
32 eliun 4958 . . . . . . . . . . . 12 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
33 nfixp1 8856 . . . . . . . . . . . . . 14 𝑥X𝑥𝐴 𝐵
3433nfel2 2925 . . . . . . . . . . . . 13 𝑥 𝑔X𝑥𝐴 𝐵
35 nfv 1917 . . . . . . . . . . . . . 14 𝑥𝑦𝐴 𝑧 = (𝑓𝑦)
3633, 35nfrexw 3296 . . . . . . . . . . . . 13 𝑥𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)
37 simplrr 776 . . . . . . . . . . . . . . . . . . . 20 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → 𝑧𝐵)
38 iftrue 4492 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) = 𝑧)
39 csbeq1a 3869 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘𝐵 = 𝑘 / 𝑥𝐵)
4039equcoms 2023 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑥𝐵 = 𝑘 / 𝑥𝐵)
4140eqcomd 2742 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑥𝑘 / 𝑥𝐵 = 𝐵)
4238, 41eleq12d 2832 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵𝑧𝐵))
4337, 42syl5ibrcom 246 . . . . . . . . . . . . . . . . . . 19 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
44 vex 3449 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑔 ∈ V
4544elixp 8842 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔X𝑥𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
4645simprbi 497 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
4746adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
48 nfv 1917 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑔𝑥) ∈ 𝐵
49 nfcsb1v 3880 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥𝑘 / 𝑥𝐵
5049nfel2 2925 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥(𝑔𝑘) ∈ 𝑘 / 𝑥𝐵
51 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑘 → (𝑔𝑥) = (𝑔𝑘))
5251, 39eleq12d 2832 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘 → ((𝑔𝑥) ∈ 𝐵 ↔ (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵))
5348, 50, 52cbvralw 3289 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵 ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
5447, 53sylib 217 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
5554r19.21bi 3234 . . . . . . . . . . . . . . . . . . . 20 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
56 iffalse 4495 . . . . . . . . . . . . . . . . . . . . 21 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) = (𝑔𝑘))
5756eleq1d 2822 . . . . . . . . . . . . . . . . . . . 20 𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵 ↔ (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵))
5855, 57syl5ibrcom 246 . . . . . . . . . . . . . . . . . . 19 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (¬ 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
5943, 58pm2.61d 179 . . . . . . . . . . . . . . . . . 18 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵)
6059ralrimiva 3143 . . . . . . . . . . . . . . . . 17 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵)
61 ixpfn 8841 . . . . . . . . . . . . . . . . . . . . 21 (𝑔X𝑥𝐴 𝐵𝑔 Fn 𝐴)
6261adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑔 Fn 𝐴)
6362fndmd 6607 . . . . . . . . . . . . . . . . . . 19 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → dom 𝑔 = 𝐴)
6444dmex 7848 . . . . . . . . . . . . . . . . . . 19 dom 𝑔 ∈ V
6563, 64eqeltrrdi 2847 . . . . . . . . . . . . . . . . . 18 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝐴 ∈ V)
66 mptelixpg 8873 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ V → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑘𝐴 𝑘 / 𝑥𝐵 ↔ ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
6765, 66syl 17 . . . . . . . . . . . . . . . . 17 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑘𝐴 𝑘 / 𝑥𝐵 ↔ ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
6860, 67mpbird 256 . . . . . . . . . . . . . . . 16 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑘𝐴 𝑘 / 𝑥𝐵)
69 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑘𝐵
7069, 49, 39cbvixp 8852 . . . . . . . . . . . . . . . 16 X𝑥𝐴 𝐵 = X𝑘𝐴 𝑘 / 𝑥𝐵
7168, 70eleqtrrdi 2849 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑥𝐴 𝐵)
72 simprl 769 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑥𝐴)
73 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))
74 vex 3449 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
7538, 73, 74fvmpt 6948 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥) = 𝑧)
7675ad2antrl 726 . . . . . . . . . . . . . . . 16 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥) = 𝑧)
7776eqcomd 2742 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥))
78 fveq1 6841 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) → (𝑓𝑦) = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦))
7978eqeq2d 2747 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) → (𝑧 = (𝑓𝑦) ↔ 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦)))
80 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦) = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥))
8180eqeq2d 2747 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦) ↔ 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥)))
8279, 81rspc2ev 3592 . . . . . . . . . . . . . . 15 (((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑥𝐴 𝐵𝑥𝐴𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥)) → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))
8371, 72, 77, 82syl3anc 1371 . . . . . . . . . . . . . 14 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))
8483exp32 421 . . . . . . . . . . . . 13 (𝑔X𝑥𝐴 𝐵 → (𝑥𝐴 → (𝑧𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))))
8534, 36, 84rexlimd 3249 . . . . . . . . . . . 12 (𝑔X𝑥𝐴 𝐵 → (∃𝑥𝐴 𝑧𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
8632, 85biimtrid 241 . . . . . . . . . . 11 (𝑔X𝑥𝐴 𝐵 → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
8786exlimiv 1933 . . . . . . . . . 10 (∃𝑔 𝑔X𝑥𝐴 𝐵 → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
8831, 87sylbi 216 . . . . . . . . 9 (X𝑥𝐴 𝐵 ≠ ∅ → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
89883ad2ant3 1135 . . . . . . . 8 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9089alrimiv 1930 . . . . . . 7 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ∀𝑧(𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
91 ssab 4018 . . . . . . 7 ( 𝑥𝐴 𝐵 ⊆ {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)} ↔ ∀𝑧(𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9290, 91sylibr 233 . . . . . 6 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 ⊆ {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)})
9317rnmpo 7489 . . . . . 6 ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) = {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)}
9492, 93sseqtrrdi 3995 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 ⊆ ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
9519frnd 6676 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ⊆ 𝑥𝐴 𝐵)
9694, 95eqssd 3961 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 = ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
97 foeq3 6754 . . . 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))))
9896, 97syl 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))))
9930, 98mpbird 256 . 2 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto 𝑥𝐴 𝐵)
100 fowdom 9507 . 2 (((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ∈ V ∧ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵* (X𝑥𝐴 𝐵 × 𝐴))
10127, 99, 100syl2anc 584 1 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵* (X𝑥𝐴 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 2713  wne 2943  wral 3064  wrex 3073  Vcvv 3445  csb 3855  wss 3910  c0 4282  ifcif 4486   ciun 4954   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  Xcixp 8835  * cwdom 9500
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-ixp 8836  df-wdom 9501
This theorem is referenced by:  ptcmplem2  23404
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