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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjf1o Structured version   Visualization version   GIF version
Theorem disjf1o 43875
Description: A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
disjf1o.xph β„²π‘₯πœ‘
disjf1o.f 𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡)
disjf1o.b ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)
disjf1o.dj (πœ‘ β†’ Disj π‘₯ ∈ 𝐴 𝐡)
disjf1o.d 𝐢 = {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  βˆ…}
disjf1o.e 𝐷 = (ran 𝐹 βˆ– {βˆ…})
Assertion
Ref Expression
disjf1o (πœ‘ β†’ (𝐹 β†Ύ 𝐢):𝐢–1-1-onto→𝐷)
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐢   π‘₯,𝐷   π‘₯,𝑉
Allowed substitution hints:   πœ‘(π‘₯)   𝐡(π‘₯)   𝐹(π‘₯)
Proof of Theorem disjf1o
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 disjf1o.xph . . . 4 β„²π‘₯πœ‘
2 eqid 2733 . . . 4 (π‘₯ ∈ 𝐢 ↦ 𝐡) = (π‘₯ ∈ 𝐢 ↦ 𝐡)
3 simpl 484 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ πœ‘)
4 disjf1o.d . . . . . . . 8 𝐢 = {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  βˆ…}
5 ssrab2 4077 . . . . . . . 8 {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  βˆ…} βŠ† 𝐴
64, 5eqsstri 4016 . . . . . . 7 𝐢 βŠ† 𝐴
7 id 22 . . . . . . 7 (π‘₯ ∈ 𝐢 β†’ π‘₯ ∈ 𝐢)
86, 7sselid 3980 . . . . . 6 (π‘₯ ∈ 𝐢 β†’ π‘₯ ∈ 𝐴)
98adantl 483 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ 𝐴)
10 disjf1o.b . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ 𝑉)
113, 9, 10syl2anc 585 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐡 ∈ 𝑉)
127, 4eleqtrdi 2844 . . . . . . 7 (π‘₯ ∈ 𝐢 β†’ π‘₯ ∈ {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  βˆ…})
13 rabid 3453 . . . . . . . 8 (π‘₯ ∈ {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  βˆ…} ↔ (π‘₯ ∈ 𝐴 ∧ 𝐡 β‰  βˆ…))
1413a1i 11 . . . . . . 7 (π‘₯ ∈ 𝐢 β†’ (π‘₯ ∈ {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  βˆ…} ↔ (π‘₯ ∈ 𝐴 ∧ 𝐡 β‰  βˆ…)))
1512, 14mpbid 231 . . . . . 6 (π‘₯ ∈ 𝐢 β†’ (π‘₯ ∈ 𝐴 ∧ 𝐡 β‰  βˆ…))
1615simprd 497 . . . . 5 (π‘₯ ∈ 𝐢 β†’ 𝐡 β‰  βˆ…)
1716adantl 483 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐡 β‰  βˆ…)
186a1i 11 . . . . 5 (πœ‘ β†’ 𝐢 βŠ† 𝐴)
19 disjf1o.dj . . . . 5 (πœ‘ β†’ Disj π‘₯ ∈ 𝐴 𝐡)
20 disjss1 5119 . . . . 5 (𝐢 βŠ† 𝐴 β†’ (Disj π‘₯ ∈ 𝐴 𝐡 β†’ Disj π‘₯ ∈ 𝐢 𝐡))
2118, 19, 20sylc 65 . . . 4 (πœ‘ β†’ Disj π‘₯ ∈ 𝐢 𝐡)
221, 2, 11, 17, 21disjf1 43866 . . 3 (πœ‘ β†’ (π‘₯ ∈ 𝐢 ↦ 𝐡):𝐢–1-1→𝑉)
23 f1f1orn 6842 . . 3 ((π‘₯ ∈ 𝐢 ↦ 𝐡):𝐢–1-1→𝑉 β†’ (π‘₯ ∈ 𝐢 ↦ 𝐡):𝐢–1-1-ontoβ†’ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
2422, 23syl 17 . 2 (πœ‘ β†’ (π‘₯ ∈ 𝐢 ↦ 𝐡):𝐢–1-1-ontoβ†’ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
25 disjf1o.f . . . . . 6 𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡)
2625a1i 11 . . . . 5 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡))
2726reseq1d 5979 . . . 4 (πœ‘ β†’ (𝐹 β†Ύ 𝐢) = ((π‘₯ ∈ 𝐴 ↦ 𝐡) β†Ύ 𝐢))
2818resmptd 6039 . . . 4 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐡) β†Ύ 𝐢) = (π‘₯ ∈ 𝐢 ↦ 𝐡))
2927, 28eqtrd 2773 . . 3 (πœ‘ β†’ (𝐹 β†Ύ 𝐢) = (π‘₯ ∈ 𝐢 ↦ 𝐡))
30 eqidd 2734 . . 3 (πœ‘ β†’ 𝐢 = 𝐢)
31 simpl 484 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐷) β†’ πœ‘)
32 id 22 . . . . . . . . . 10 (𝑦 ∈ 𝐷 β†’ 𝑦 ∈ 𝐷)
33 disjf1o.e . . . . . . . . . 10 𝐷 = (ran 𝐹 βˆ– {βˆ…})
3432, 33eleqtrdi 2844 . . . . . . . . 9 (𝑦 ∈ 𝐷 β†’ 𝑦 ∈ (ran 𝐹 βˆ– {βˆ…}))
35 eldifsni 4793 . . . . . . . . 9 (𝑦 ∈ (ran 𝐹 βˆ– {βˆ…}) β†’ 𝑦 β‰  βˆ…)
3634, 35syl 17 . . . . . . . 8 (𝑦 ∈ 𝐷 β†’ 𝑦 β‰  βˆ…)
3736adantl 483 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐷) β†’ 𝑦 β‰  βˆ…)
38 eldifi 4126 . . . . . . . . . 10 (𝑦 ∈ (ran 𝐹 βˆ– {βˆ…}) β†’ 𝑦 ∈ ran 𝐹)
3934, 38syl 17 . . . . . . . . 9 (𝑦 ∈ 𝐷 β†’ 𝑦 ∈ ran 𝐹)
4025elrnmpt 5954 . . . . . . . . . 10 (𝑦 ∈ ran 𝐹 β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡))
4139, 40syl 17 . . . . . . . . 9 (𝑦 ∈ 𝐷 β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡))
4239, 41mpbid 231 . . . . . . . 8 (𝑦 ∈ 𝐷 β†’ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡)
4342adantl 483 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐷) β†’ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡)
44 nfv 1918 . . . . . . . . . 10 β„²π‘₯ 𝑦 β‰  βˆ…
451, 44nfan 1903 . . . . . . . . 9 β„²π‘₯(πœ‘ ∧ 𝑦 β‰  βˆ…)
46 nfcv 2904 . . . . . . . . . 10 β„²π‘₯𝑦
47 nfmpt1 5256 . . . . . . . . . . 11 β„²π‘₯(π‘₯ ∈ 𝐢 ↦ 𝐡)
4847nfrn 5950 . . . . . . . . . 10 β„²π‘₯ran (π‘₯ ∈ 𝐢 ↦ 𝐡)
4946, 48nfel 2918 . . . . . . . . 9 β„²π‘₯ 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡)
50 simp3 1139 . . . . . . . . . . . 12 ((𝑦 β‰  βˆ… ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ 𝑦 = 𝐡)
51 simp2 1138 . . . . . . . . . . . . . . . 16 ((𝑦 β‰  βˆ… ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ π‘₯ ∈ 𝐴)
52 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐡 β†’ 𝑦 = 𝐡)
5352eqcomd 2739 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐡 β†’ 𝐡 = 𝑦)
5453adantl 483 . . . . . . . . . . . . . . . . . 18 ((𝑦 β‰  βˆ… ∧ 𝑦 = 𝐡) β†’ 𝐡 = 𝑦)
55 simpl 484 . . . . . . . . . . . . . . . . . 18 ((𝑦 β‰  βˆ… ∧ 𝑦 = 𝐡) β†’ 𝑦 β‰  βˆ…)
5654, 55eqnetrd 3009 . . . . . . . . . . . . . . . . 17 ((𝑦 β‰  βˆ… ∧ 𝑦 = 𝐡) β†’ 𝐡 β‰  βˆ…)
57563adant2 1132 . . . . . . . . . . . . . . . 16 ((𝑦 β‰  βˆ… ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ 𝐡 β‰  βˆ…)
5851, 57jca 513 . . . . . . . . . . . . . . 15 ((𝑦 β‰  βˆ… ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ (π‘₯ ∈ 𝐴 ∧ 𝐡 β‰  βˆ…))
5958, 13sylibr 233 . . . . . . . . . . . . . 14 ((𝑦 β‰  βˆ… ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ π‘₯ ∈ {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  βˆ…})
604eqcomi 2742 . . . . . . . . . . . . . . 15 {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  βˆ…} = 𝐢
6160a1i 11 . . . . . . . . . . . . . 14 ((𝑦 β‰  βˆ… ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  βˆ…} = 𝐢)
6259, 61eleqtrd 2836 . . . . . . . . . . . . 13 ((𝑦 β‰  βˆ… ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ π‘₯ ∈ 𝐢)
63 eqvisset 3492 . . . . . . . . . . . . . 14 (𝑦 = 𝐡 β†’ 𝐡 ∈ V)
64633ad2ant3 1136 . . . . . . . . . . . . 13 ((𝑦 β‰  βˆ… ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ 𝐡 ∈ V)
652elrnmpt1 5956 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝐢 ∧ 𝐡 ∈ V) β†’ 𝐡 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
6662, 64, 65syl2anc 585 . . . . . . . . . . . 12 ((𝑦 β‰  βˆ… ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ 𝐡 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
6750, 66eqeltrd 2834 . . . . . . . . . . 11 ((𝑦 β‰  βˆ… ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
68673adant1l 1177 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 β‰  βˆ…) ∧ π‘₯ ∈ 𝐴 ∧ 𝑦 = 𝐡) β†’ 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
69683exp 1120 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 β‰  βˆ…) β†’ (π‘₯ ∈ 𝐴 β†’ (𝑦 = 𝐡 β†’ 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡))))
7045, 49, 69rexlimd 3264 . . . . . . . 8 ((πœ‘ ∧ 𝑦 β‰  βˆ…) β†’ (βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡 β†’ 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡)))
7170imp 408 . . . . . . 7 (((πœ‘ ∧ 𝑦 β‰  βˆ…) ∧ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡) β†’ 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
7231, 37, 43, 71syl21anc 837 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐷) β†’ 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
7372ralrimiva 3147 . . . . 5 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐷 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
74 dfss3 3970 . . . . 5 (𝐷 βŠ† ran (π‘₯ ∈ 𝐢 ↦ 𝐡) ↔ βˆ€π‘¦ ∈ 𝐷 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
7573, 74sylibr 233 . . . 4 (πœ‘ β†’ 𝐷 βŠ† ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
76 simpl 484 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡)) β†’ πœ‘)
77 vex 3479 . . . . . . . 8 𝑦 ∈ V
782elrnmpt 5954 . . . . . . . 8 (𝑦 ∈ V β†’ (𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡) ↔ βˆƒπ‘₯ ∈ 𝐢 𝑦 = 𝐡))
7977, 78ax-mp 5 . . . . . . 7 (𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡) ↔ βˆƒπ‘₯ ∈ 𝐢 𝑦 = 𝐡)
8079biimpi 215 . . . . . 6 (𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡) β†’ βˆƒπ‘₯ ∈ 𝐢 𝑦 = 𝐡)
8180adantl 483 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡)) β†’ βˆƒπ‘₯ ∈ 𝐢 𝑦 = 𝐡)
82 nfv 1918 . . . . . . 7 β„²π‘₯ 𝑦 ∈ 𝐷
83 simpr 486 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ 𝑦 = 𝐡)
848adantr 482 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ π‘₯ ∈ 𝐴)
8583, 63syl 17 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ 𝐡 ∈ V)
8625elrnmpt1 5956 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝐴 ∧ 𝐡 ∈ V) β†’ 𝐡 ∈ ran 𝐹)
8784, 85, 86syl2anc 585 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ 𝐡 ∈ ran 𝐹)
8883, 87eqeltrd 2834 . . . . . . . . . . 11 ((π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ 𝑦 ∈ ran 𝐹)
89883adant1 1131 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ 𝑦 ∈ ran 𝐹)
9016adantr 482 . . . . . . . . . . . . 13 ((π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ 𝐡 β‰  βˆ…)
9183, 90eqnetrd 3009 . . . . . . . . . . . 12 ((π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ 𝑦 β‰  βˆ…)
92 nelsn 4668 . . . . . . . . . . . 12 (𝑦 β‰  βˆ… β†’ Β¬ 𝑦 ∈ {βˆ…})
9391, 92syl 17 . . . . . . . . . . 11 ((π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ Β¬ 𝑦 ∈ {βˆ…})
94933adant1 1131 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ Β¬ 𝑦 ∈ {βˆ…})
9589, 94eldifd 3959 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ 𝑦 ∈ (ran 𝐹 βˆ– {βˆ…}))
9695, 33eleqtrrdi 2845 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐢 ∧ 𝑦 = 𝐡) β†’ 𝑦 ∈ 𝐷)
97963exp 1120 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝐢 β†’ (𝑦 = 𝐡 β†’ 𝑦 ∈ 𝐷)))
981, 82, 97rexlimd 3264 . . . . . 6 (πœ‘ β†’ (βˆƒπ‘₯ ∈ 𝐢 𝑦 = 𝐡 β†’ 𝑦 ∈ 𝐷))
9998imp 408 . . . . 5 ((πœ‘ ∧ βˆƒπ‘₯ ∈ 𝐢 𝑦 = 𝐡) β†’ 𝑦 ∈ 𝐷)
10076, 81, 99syl2anc 585 . . . 4 ((πœ‘ ∧ 𝑦 ∈ ran (π‘₯ ∈ 𝐢 ↦ 𝐡)) β†’ 𝑦 ∈ 𝐷)
10175, 100eqelssd 4003 . . 3 (πœ‘ β†’ 𝐷 = ran (π‘₯ ∈ 𝐢 ↦ 𝐡))
10229, 30, 101f1oeq123d 6825 . 2 (πœ‘ β†’ ((𝐹 β†Ύ 𝐢):𝐢–1-1-onto→𝐷 ↔ (π‘₯ ∈ 𝐢 ↦ 𝐡):𝐢–1-1-ontoβ†’ran (π‘₯ ∈ 𝐢 ↦ 𝐡)))
10324, 102mpbird 257 1 (πœ‘ β†’ (𝐹 β†Ύ 𝐢):𝐢–1-1-onto→𝐷)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  β„²wnf 1786   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βˆ– cdif 3945   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  Disj wdisj 5113   ↦ cmpt 5231  ran crn 5677   β†Ύ cres 5678  β€“1-1β†’wf1 6538  β€“1-1-ontoβ†’wf1o 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549
This theorem is referenced by:  sge0fodjrnlem  45119
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