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Theorem ixpiunwdom 8661
Description: Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8104 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 3343 . . . . . . . . . 10 𝑓 ∈ V
21elixp 8081 . . . . . . . . 9 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simprbi 483 . . . . . . . 8 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
4 ssiun2 4715 . . . . . . . . . 10 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
54sseld 3743 . . . . . . . . 9 (𝑥𝐴 → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ 𝑥𝐴 𝐵))
65ralimia 3088 . . . . . . . 8 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵)
73, 6syl 17 . . . . . . 7 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵)
8 nfv 1992 . . . . . . . 8 𝑦(𝑓𝑥) ∈ 𝑥𝐴 𝐵
9 nfiu1 4702 . . . . . . . . 9 𝑥 𝑥𝐴 𝐵
109nfel2 2919 . . . . . . . 8 𝑥(𝑓𝑦) ∈ 𝑥𝐴 𝐵
11 fveq2 6352 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
1211eleq1d 2824 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝑥𝐴 𝐵 ↔ (𝑓𝑦) ∈ 𝑥𝐴 𝐵))
138, 10, 12cbvral 3306 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
147, 13sylib 208 . . . . . 6 (𝑓X𝑥𝐴 𝐵 → ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
1514adantl 473 . . . . 5 (((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) ∧ 𝑓X𝑥𝐴 𝐵) → ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
1615ralrimiva 3104 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ∀𝑓X 𝑥𝐴 𝐵𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
17 eqid 2760 . . . . 5 (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) = (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦))
1817fmpt2 7405 . . . 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 8103 . . . . . 6 ( 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
21203ad2ant2 1129 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
22 ovex 6841 . . . . . 6 ( 𝑥𝐴 𝐵𝑚 𝐴) ∈ V
2322ssex 4954 . . . . 5 (X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴) → X𝑥𝐴 𝐵 ∈ V)
2421, 23syl 17 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → X𝑥𝐴 𝐵 ∈ V)
25 simp1 1131 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝐴𝑉)
26 xpexg 7125 . . . 4 ((X𝑥𝐴 𝐵 ∈ V ∧ 𝐴𝑉) → (X𝑥𝐴 𝐵 × 𝐴) ∈ V)
2724, 25, 26syl2anc 696 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (X𝑥𝐴 𝐵 × 𝐴) ∈ V)
28 simp2 1132 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵𝑊)
29 fex2 7286 . . 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 1477 . 2 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ∈ V)
31 ffn 6206 . . . . 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 6282 . . . 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 4074 . . . . . . . . . 10 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑔 𝑔X𝑥𝐴 𝐵)
36 eliun 4676 . . . . . . . . . . . 12 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
37 nfixp1 8094 . . . . . . . . . . . . . 14 𝑥X𝑥𝐴 𝐵
3837nfel2 2919 . . . . . . . . . . . . 13 𝑥 𝑔X𝑥𝐴 𝐵
39 nfv 1992 . . . . . . . . . . . . . 14 𝑥𝑦𝐴 𝑧 = (𝑓𝑦)
4037, 39nfrex 3145 . . . . . . . . . . . . 13 𝑥𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)
41 simplrr 820 . . . . . . . . . . . . . . . . . . . 20 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → 𝑧𝐵)
42 iftrue 4236 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) = 𝑧)
43 csbeq1a 3683 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘𝐵 = 𝑘 / 𝑥𝐵)
4443equcoms 2102 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑥𝐵 = 𝑘 / 𝑥𝐵)
4544eqcomd 2766 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑥𝑘 / 𝑥𝐵 = 𝐵)
4642, 45eleq12d 2833 . . . . . . . . . . . . . . . . . . . 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 3343 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑔 ∈ V
4948elixp 8081 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔X𝑥𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
5049simprbi 483 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
5150adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
52 nfv 1992 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑔𝑥) ∈ 𝐵
53 nfcsb1v 3690 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥𝑘 / 𝑥𝐵
5453nfel2 2919 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥(𝑔𝑘) ∈ 𝑘 / 𝑥𝐵
55 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑘 → (𝑔𝑥) = (𝑔𝑘))
5655, 43eleq12d 2833 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘 → ((𝑔𝑥) ∈ 𝐵 ↔ (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵))
5752, 54, 56cbvral 3306 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵 ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
5851, 57sylib 208 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
5958r19.21bi 3070 . . . . . . . . . . . . . . . . . . . 20 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
60 iffalse 4239 . . . . . . . . . . . . . . . . . . . . 21 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) = (𝑔𝑘))
6160eleq1d 2824 . . . . . . . . . . . . . . . . . . . 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 3104 . . . . . . . . . . . . . . . . 17 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵)
65 ixpfn 8080 . . . . . . . . . . . . . . . . . . . . 21 (𝑔X𝑥𝐴 𝐵𝑔 Fn 𝐴)
6665adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑔 Fn 𝐴)
67 fndm 6151 . . . . . . . . . . . . . . . . . . . 20 (𝑔 Fn 𝐴 → dom 𝑔 = 𝐴)
6866, 67syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → dom 𝑔 = 𝐴)
6948dmex 7264 . . . . . . . . . . . . . . . . . . 19 dom 𝑔 ∈ V
7068, 69syl6eqelr 2848 . . . . . . . . . . . . . . . . . 18 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝐴 ∈ V)
71 mptelixpg 8111 . . . . . . . . . . . . . . . . . 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 2902 . . . . . . . . . . . . . . . . 17 𝑘𝐵
7574, 53, 43cbvixp 8091 . . . . . . . . . . . . . . . 16 X𝑥𝐴 𝐵 = X𝑘𝐴 𝑘 / 𝑥𝐵
7673, 75syl6eleqr 2850 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑥𝐴 𝐵)
77 simprl 811 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑥𝐴)
78 eqid 2760 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))
79 vex 3343 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
8042, 78, 79fvmpt 6444 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥) = 𝑧)
8180ad2antrl 766 . . . . . . . . . . . . . . . 16 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥) = 𝑧)
8281eqcomd 2766 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥))
83 fveq1 6351 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) → (𝑓𝑦) = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦))
8483eqeq2d 2770 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) → (𝑧 = (𝑓𝑦) ↔ 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦)))
85 fveq2 6352 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦) = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥))
8685eqeq2d 2770 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦) ↔ 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥)))
8784, 86rspc2ev 3463 . . . . . . . . . . . . . . 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 1477 . . . . . . . . . . . . . 14 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))
8988exp32 632 . . . . . . . . . . . . 13 (𝑔X𝑥𝐴 𝐵 → (𝑥𝐴 → (𝑧𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))))
9038, 40, 89rexlimd 3164 . . . . . . . . . . . 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 2007 . . . . . . . . . 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 1130 . . . . . . . 8 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9594alrimiv 2004 . . . . . . 7 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ∀𝑧(𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
96 ssab 3813 . . . . . . 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 6935 . . . . . 6 ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) = {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)}
9997, 98syl6sseqr 3793 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 ⊆ ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
100 frn 6214 . . . . . 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 3761 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 = ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
103 foeq3 6274 . . . 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 8641 . 2 (((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ∈ V ∧ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵* (X𝑥𝐴 𝐵 × 𝐴))
10730, 105, 106syl2anc 696 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 383  w3a 1072  wal 1630   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  Vcvv 3340  csb 3674  wss 3715  c0 4058  ifcif 4230   ciun 4672   class class class wbr 4804  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267   Fn wfn 6044  wf 6045  ontowfo 6047  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023  Xcixp 8074  * cwdom 8627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-ixp 8075  df-wdom 8629
This theorem is referenced by:  ptcmplem2  22058
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