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Theorem ixpiunwdom 8612
 Description: Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8055 this shows that ∪ 𝑥 ∈ 𝐴𝐵 and X𝑥 ∈ 𝐴𝐵 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 3307 . . . . . . . . . 10 𝑓 ∈ V
21elixp 8032 . . . . . . . . 9 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simprbi 483 . . . . . . . 8 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
4 ssiun2 4671 . . . . . . . . . 10 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
54sseld 3708 . . . . . . . . 9 (𝑥𝐴 → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ 𝑥𝐴 𝐵))
65ralimia 3052 . . . . . . . 8 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵)
73, 6syl 17 . . . . . . 7 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵)
8 nfv 1956 . . . . . . . 8 𝑦(𝑓𝑥) ∈ 𝑥𝐴 𝐵
9 nfiu1 4658 . . . . . . . . 9 𝑥 𝑥𝐴 𝐵
109nfel2 2883 . . . . . . . 8 𝑥(𝑓𝑦) ∈ 𝑥𝐴 𝐵
11 fveq2 6304 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
1211eleq1d 2788 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝑥𝐴 𝐵 ↔ (𝑓𝑦) ∈ 𝑥𝐴 𝐵))
138, 10, 12cbvral 3270 . . . . . . 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 3068 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ∀𝑓X 𝑥𝐴 𝐵𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
17 eqid 2724 . . . . 5 (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) = (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦))
1817fmpt2 7357 . . . 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 8054 . . . . . 6 ( 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
21203ad2ant2 1126 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
22 ovex 6793 . . . . . 6 ( 𝑥𝐴 𝐵𝑚 𝐴) ∈ V
2322ssex 4910 . . . . 5 (X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴) → X𝑥𝐴 𝐵 ∈ V)
2421, 23syl 17 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → X𝑥𝐴 𝐵 ∈ V)
25 simp1 1128 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝐴𝑉)
26 xpexg 7077 . . . 4 ((X𝑥𝐴 𝐵 ∈ V ∧ 𝐴𝑉) → (X𝑥𝐴 𝐵 × 𝐴) ∈ V)
2724, 25, 26syl2anc 696 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (X𝑥𝐴 𝐵 × 𝐴) ∈ V)
28 simp2 1129 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵𝑊)
29 fex2 7238 . . 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 1439 . 2 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ∈ V)
31 ffn 6158 . . . . 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 6234 . . . 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 4039 . . . . . . . . . 10 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑔 𝑔X𝑥𝐴 𝐵)
36 eliun 4632 . . . . . . . . . . . 12 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
37 nfixp1 8045 . . . . . . . . . . . . . 14 𝑥X𝑥𝐴 𝐵
3837nfel2 2883 . . . . . . . . . . . . 13 𝑥 𝑔X𝑥𝐴 𝐵
39 nfv 1956 . . . . . . . . . . . . . 14 𝑥𝑦𝐴 𝑧 = (𝑓𝑦)
4037, 39nfrex 3109 . . . . . . . . . . . . 13 𝑥𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)
41 simplrr 820 . . . . . . . . . . . . . . . . . . . 20 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → 𝑧𝐵)
42 iftrue 4200 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) = 𝑧)
43 csbeq1a 3648 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘𝐵 = 𝑘 / 𝑥𝐵)
4443equcoms 2066 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑥𝐵 = 𝑘 / 𝑥𝐵)
4544eqcomd 2730 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑥𝑘 / 𝑥𝐵 = 𝐵)
4642, 45eleq12d 2797 . . . . . . . . . . . . . . . . . . . 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 3307 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑔 ∈ V
4948elixp 8032 . . . . . . . . . . . . . . . . . . . . . . . 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 1956 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑔𝑥) ∈ 𝐵
53 nfcsb1v 3655 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥𝑘 / 𝑥𝐵
5453nfel2 2883 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥(𝑔𝑘) ∈ 𝑘 / 𝑥𝐵
55 fveq2 6304 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑘 → (𝑔𝑥) = (𝑔𝑘))
5655, 43eleq12d 2797 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘 → ((𝑔𝑥) ∈ 𝐵 ↔ (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵))
5752, 54, 56cbvral 3270 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵 ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
5851, 57sylib 208 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
5958r19.21bi 3034 . . . . . . . . . . . . . . . . . . . 20 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
60 iffalse 4203 . . . . . . . . . . . . . . . . . . . . 21 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) = (𝑔𝑘))
6160eleq1d 2788 . . . . . . . . . . . . . . . . . . . 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 3068 . . . . . . . . . . . . . . . . 17 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵)
65 ixpfn 8031 . . . . . . . . . . . . . . . . . . . . 21 (𝑔X𝑥𝐴 𝐵𝑔 Fn 𝐴)
6665adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑔 Fn 𝐴)
67 fndm 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑔 Fn 𝐴 → dom 𝑔 = 𝐴)
6866, 67syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → dom 𝑔 = 𝐴)
6948dmex 7216 . . . . . . . . . . . . . . . . . . 19 dom 𝑔 ∈ V
7068, 69syl6eqelr 2812 . . . . . . . . . . . . . . . . . 18 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝐴 ∈ V)
71 mptelixpg 8062 . . . . . . . . . . . . . . . . . 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 2866 . . . . . . . . . . . . . . . . 17 𝑘𝐵
7574, 53, 43cbvixp 8042 . . . . . . . . . . . . . . . 16 X𝑥𝐴 𝐵 = X𝑘𝐴 𝑘 / 𝑥𝐵
7673, 75syl6eleqr 2814 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑥𝐴 𝐵)
77 simprl 811 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑥𝐴)
78 eqid 2724 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))
79 vex 3307 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
8042, 78, 79fvmpt 6396 . . . . . . . . . . . . . . . . 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 2730 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥))
83 fveq1 6303 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) → (𝑓𝑦) = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦))
8483eqeq2d 2734 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) → (𝑧 = (𝑓𝑦) ↔ 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦)))
85 fveq2 6304 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦) = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥))
8685eqeq2d 2734 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦) ↔ 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥)))
8784, 86rspc2ev 3428 . . . . . . . . . . . . . . 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 1439 . . . . . . . . . . . . . 14 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))
8988exp32 632 . . . . . . . . . . . . 13 (𝑔X𝑥𝐴 𝐵 → (𝑥𝐴 → (𝑧𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))))
9038, 40, 89rexlimd 3128 . . . . . . . . . . . 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 1971 . . . . . . . . . 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 1127 . . . . . . . 8 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9594alrimiv 1968 . . . . . . 7 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ∀𝑧(𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
96 ssab 3778 . . . . . . 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 6887 . . . . . 6 ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) = {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)}
9997, 98syl6sseqr 3758 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 ⊆ ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
100 frn 6166 . . . . . 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 3726 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 = ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
103 foeq3 6226 . . . 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 8592 . 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 1594   = wceq 1596  ∃wex 1817   ∈ wcel 2103  {cab 2710   ≠ wne 2896  ∀wral 3014  ∃wrex 3015  Vcvv 3304  ⦋csb 3639   ⊆ wss 3680  ∅c0 4023  ifcif 4194  ∪ ciun 4628   class class class wbr 4760   ↦ cmpt 4837   × cxp 5216  dom cdm 5218  ran crn 5219   Fn wfn 5996  ⟶wf 5997  –onto→wfo 5999  ‘cfv 6001  (class class class)co 6765   ↦ cmpt2 6767   ↑𝑚 cmap 7974  Xcixp 8025   ≼* cwdom 8578 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-map 7976  df-ixp 8026  df-wdom 8580 This theorem is referenced by:  ptcmplem2  21979
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