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Theorem ixpiunwdom 8703
Description: Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 8143 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 3353 . . . . . . . . . 10 𝑓 ∈ V
21elixp 8120 . . . . . . . . 9 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
32simprbi 490 . . . . . . . 8 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
4 ssiun2 4719 . . . . . . . . . 10 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
54sseld 3760 . . . . . . . . 9 (𝑥𝐴 → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ 𝑥𝐴 𝐵))
65ralimia 3097 . . . . . . . 8 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵)
73, 6syl 17 . . . . . . 7 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵)
8 nfv 2009 . . . . . . . 8 𝑦(𝑓𝑥) ∈ 𝑥𝐴 𝐵
9 nfiu1 4706 . . . . . . . . 9 𝑥 𝑥𝐴 𝐵
109nfel2 2924 . . . . . . . 8 𝑥(𝑓𝑦) ∈ 𝑥𝐴 𝐵
11 fveq2 6375 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑓𝑥) = (𝑓𝑦))
1211eleq1d 2829 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑓𝑥) ∈ 𝑥𝐴 𝐵 ↔ (𝑓𝑦) ∈ 𝑥𝐴 𝐵))
138, 10, 12cbvral 3315 . . . . . . 7 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
147, 13sylib 209 . . . . . 6 (𝑓X𝑥𝐴 𝐵 → ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
1514adantl 473 . . . . 5 (((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) ∧ 𝑓X𝑥𝐴 𝐵) → ∀𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
1615ralrimiva 3113 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ∀𝑓X 𝑥𝐴 𝐵𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵)
17 eqid 2765 . . . . 5 (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) = (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦))
1817fmpt2 7438 . . . 4 (∀𝑓X 𝑥𝐴 𝐵𝑦𝐴 (𝑓𝑦) ∈ 𝑥𝐴 𝐵 ↔ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)⟶ 𝑥𝐴 𝐵)
1916, 18sylib 209 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)⟶ 𝑥𝐴 𝐵)
20 ixpssmap2g 8142 . . . . . 6 ( 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
21203ad2ant2 1164 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
22 ovex 6874 . . . . . 6 ( 𝑥𝐴 𝐵𝑚 𝐴) ∈ V
2322ssex 4963 . . . . 5 (X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴) → X𝑥𝐴 𝐵 ∈ V)
2421, 23syl 17 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → X𝑥𝐴 𝐵 ∈ V)
25 simp1 1166 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝐴𝑉)
26 xpexg 7158 . . . 4 ((X𝑥𝐴 𝐵 ∈ V ∧ 𝐴𝑉) → (X𝑥𝐴 𝐵 × 𝐴) ∈ V)
2724, 25, 26syl2anc 579 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (X𝑥𝐴 𝐵 × 𝐴) ∈ V)
28 simp2 1167 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵𝑊)
29 fex2 7319 . . 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 1490 . 2 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ∈ V)
3119ffnd 6224 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) Fn (X𝑥𝐴 𝐵 × 𝐴))
32 dffn4 6304 . . . 4 ((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) Fn (X𝑥𝐴 𝐵 × 𝐴) ↔ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto→ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
3331, 32sylib 209 . . 3 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto→ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
34 n0 4095 . . . . . . . . . 10 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑔 𝑔X𝑥𝐴 𝐵)
35 eliun 4680 . . . . . . . . . . . 12 (𝑧 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧𝐵)
36 nfixp1 8133 . . . . . . . . . . . . . 14 𝑥X𝑥𝐴 𝐵
3736nfel2 2924 . . . . . . . . . . . . 13 𝑥 𝑔X𝑥𝐴 𝐵
38 nfv 2009 . . . . . . . . . . . . . 14 𝑥𝑦𝐴 𝑧 = (𝑓𝑦)
3936, 38nfrex 3153 . . . . . . . . . . . . 13 𝑥𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)
40 simplrr 796 . . . . . . . . . . . . . . . . . . . 20 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → 𝑧𝐵)
41 iftrue 4249 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) = 𝑧)
42 csbeq1a 3700 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘𝐵 = 𝑘 / 𝑥𝐵)
4342equcoms 2117 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑥𝐵 = 𝑘 / 𝑥𝐵)
4443eqcomd 2771 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑥𝑘 / 𝑥𝐵 = 𝐵)
4541, 44eleq12d 2838 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵𝑧𝐵))
4640, 45syl5ibrcom 238 . . . . . . . . . . . . . . . . . . 19 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
47 vex 3353 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑔 ∈ V
4847elixp 8120 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔X𝑥𝐴 𝐵 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))
4948simprbi 490 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔X𝑥𝐴 𝐵 → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
5049adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵)
51 nfv 2009 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑔𝑥) ∈ 𝐵
52 nfcsb1v 3707 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑥𝑘 / 𝑥𝐵
5352nfel2 2924 . . . . . . . . . . . . . . . . . . . . . . 23 𝑥(𝑔𝑘) ∈ 𝑘 / 𝑥𝐵
54 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑘 → (𝑔𝑥) = (𝑔𝑘))
5554, 42eleq12d 2838 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑘 → ((𝑔𝑥) ∈ 𝐵 ↔ (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵))
5651, 53, 55cbvral 3315 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵 ↔ ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
5750, 56sylib 209 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑘𝐴 (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
5857r19.21bi 3079 . . . . . . . . . . . . . . . . . . . 20 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵)
59 iffalse 4252 . . . . . . . . . . . . . . . . . . . . 21 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) = (𝑔𝑘))
6059eleq1d 2829 . . . . . . . . . . . . . . . . . . . 20 𝑘 = 𝑥 → (if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵 ↔ (𝑔𝑘) ∈ 𝑘 / 𝑥𝐵))
6158, 60syl5ibrcom 238 . . . . . . . . . . . . . . . . . . 19 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → (¬ 𝑘 = 𝑥 → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
6246, 61pm2.61d 171 . . . . . . . . . . . . . . . . . 18 (((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) ∧ 𝑘𝐴) → if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵)
6362ralrimiva 3113 . . . . . . . . . . . . . . . . 17 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵)
64 ixpfn 8119 . . . . . . . . . . . . . . . . . . . . 21 (𝑔X𝑥𝐴 𝐵𝑔 Fn 𝐴)
6564adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑔 Fn 𝐴)
66 fndm 6168 . . . . . . . . . . . . . . . . . . . 20 (𝑔 Fn 𝐴 → dom 𝑔 = 𝐴)
6765, 66syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → dom 𝑔 = 𝐴)
6847dmex 7297 . . . . . . . . . . . . . . . . . . 19 dom 𝑔 ∈ V
6967, 68syl6eqelr 2853 . . . . . . . . . . . . . . . . . 18 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝐴 ∈ V)
70 mptelixpg 8150 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ V → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑘𝐴 𝑘 / 𝑥𝐵 ↔ ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
7169, 70syl 17 . . . . . . . . . . . . . . . . 17 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑘𝐴 𝑘 / 𝑥𝐵 ↔ ∀𝑘𝐴 if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)) ∈ 𝑘 / 𝑥𝐵))
7263, 71mpbird 248 . . . . . . . . . . . . . . . 16 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑘𝐴 𝑘 / 𝑥𝐵)
73 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑘𝐵
7473, 52, 42cbvixp 8130 . . . . . . . . . . . . . . . 16 X𝑥𝐴 𝐵 = X𝑘𝐴 𝑘 / 𝑥𝐵
7572, 74syl6eleqr 2855 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑥𝐴 𝐵)
76 simprl 787 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑥𝐴)
77 eqid 2765 . . . . . . . . . . . . . . . . . 18 (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))
78 vex 3353 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
7941, 77, 78fvmpt 6471 . . . . . . . . . . . . . . . . 17 (𝑥𝐴 → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥) = 𝑧)
8079ad2antrl 719 . . . . . . . . . . . . . . . 16 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥) = 𝑧)
8180eqcomd 2771 . . . . . . . . . . . . . . 15 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥))
82 fveq1 6374 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) → (𝑓𝑦) = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦))
8382eqeq2d 2775 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) → (𝑧 = (𝑓𝑦) ↔ 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦)))
84 fveq2 6375 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦) = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥))
8584eqeq2d 2775 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑦) ↔ 𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥)))
8683, 85rspc2ev 3476 . . . . . . . . . . . . . . 15 (((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘))) ∈ X𝑥𝐴 𝐵𝑥𝐴𝑧 = ((𝑘𝐴 ↦ if(𝑘 = 𝑥, 𝑧, (𝑔𝑘)))‘𝑥)) → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))
8775, 76, 81, 86syl3anc 1490 . . . . . . . . . . . . . 14 ((𝑔X𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑧𝐵)) → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))
8887exp32 411 . . . . . . . . . . . . 13 (𝑔X𝑥𝐴 𝐵 → (𝑥𝐴 → (𝑧𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦))))
8937, 39, 88rexlimd 3173 . . . . . . . . . . . 12 (𝑔X𝑥𝐴 𝐵 → (∃𝑥𝐴 𝑧𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9035, 89syl5bi 233 . . . . . . . . . . 11 (𝑔X𝑥𝐴 𝐵 → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9190exlimiv 2025 . . . . . . . . . 10 (∃𝑔 𝑔X𝑥𝐴 𝐵 → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9234, 91sylbi 208 . . . . . . . . 9 (X𝑥𝐴 𝐵 ≠ ∅ → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
93923ad2ant3 1165 . . . . . . . 8 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9493alrimiv 2022 . . . . . . 7 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ∀𝑧(𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
95 ssab 3832 . . . . . . 7 ( 𝑥𝐴 𝐵 ⊆ {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)} ↔ ∀𝑧(𝑧 𝑥𝐴 𝐵 → ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)))
9694, 95sylibr 225 . . . . . 6 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 ⊆ {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)})
9717rnmpt2 6968 . . . . . 6 ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) = {𝑧 ∣ ∃𝑓X 𝑥𝐴 𝐵𝑦𝐴 𝑧 = (𝑓𝑦)}
9896, 97syl6sseqr 3812 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 ⊆ ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
9919frnd 6230 . . . . 5 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ⊆ 𝑥𝐴 𝐵)
10098, 99eqssd 3778 . . . 4 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵 = ran (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)))
101 foeq3 6296 . . . 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))))
102100, 101syl 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))))
10333, 102mpbird 248 . 2 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto 𝑥𝐴 𝐵)
104 fowdom 8683 . 2 (((𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)) ∈ V ∧ (𝑓X𝑥𝐴 𝐵, 𝑦𝐴 ↦ (𝑓𝑦)):(X𝑥𝐴 𝐵 × 𝐴)–onto 𝑥𝐴 𝐵) → 𝑥𝐴 𝐵* (X𝑥𝐴 𝐵 × 𝐴))
10530, 103, 104syl2anc 579 1 ((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵* (X𝑥𝐴 𝐵 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  Vcvv 3350  csb 3691  wss 3732  c0 4079  ifcif 4243   ciun 4676   class class class wbr 4809  cmpt 4888   × cxp 5275  dom cdm 5277  ran crn 5278   Fn wfn 6063  wf 6064  ontowfo 6066  cfv 6068  (class class class)co 6842  cmpt2 6844  𝑚 cmap 8060  Xcixp 8113  * cwdom 8669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062  df-ixp 8114  df-wdom 8671
This theorem is referenced by:  ptcmplem2  22136
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