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Theorem disjf1o 40025
Description: A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
disjf1o.xph 𝑥𝜑
disjf1o.f 𝐹 = (𝑥𝐴𝐵)
disjf1o.b ((𝜑𝑥𝐴) → 𝐵𝑉)
disjf1o.dj (𝜑Disj 𝑥𝐴 𝐵)
disjf1o.d 𝐶 = {𝑥𝐴𝐵 ≠ ∅}
disjf1o.e 𝐷 = (ran 𝐹 ∖ {∅})
Assertion
Ref Expression
disjf1o (𝜑 → (𝐹𝐶):𝐶1-1-onto𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem disjf1o
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 disjf1o.xph . . . 4 𝑥𝜑
2 eqid 2765 . . . 4 (𝑥𝐶𝐵) = (𝑥𝐶𝐵)
3 simpl 474 . . . . 5 ((𝜑𝑥𝐶) → 𝜑)
4 disjf1o.d . . . . . . . 8 𝐶 = {𝑥𝐴𝐵 ≠ ∅}
5 ssrab2 3847 . . . . . . . 8 {𝑥𝐴𝐵 ≠ ∅} ⊆ 𝐴
64, 5eqsstri 3795 . . . . . . 7 𝐶𝐴
7 id 22 . . . . . . 7 (𝑥𝐶𝑥𝐶)
86, 7sseldi 3759 . . . . . 6 (𝑥𝐶𝑥𝐴)
98adantl 473 . . . . 5 ((𝜑𝑥𝐶) → 𝑥𝐴)
10 disjf1o.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
113, 9, 10syl2anc 579 . . . 4 ((𝜑𝑥𝐶) → 𝐵𝑉)
127, 4syl6eleq 2854 . . . . . . 7 (𝑥𝐶𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅})
13 rabid 3263 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅} ↔ (𝑥𝐴𝐵 ≠ ∅))
1413a1i 11 . . . . . . 7 (𝑥𝐶 → (𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅} ↔ (𝑥𝐴𝐵 ≠ ∅)))
1512, 14mpbid 223 . . . . . 6 (𝑥𝐶 → (𝑥𝐴𝐵 ≠ ∅))
1615simprd 489 . . . . 5 (𝑥𝐶𝐵 ≠ ∅)
1716adantl 473 . . . 4 ((𝜑𝑥𝐶) → 𝐵 ≠ ∅)
186a1i 11 . . . . 5 (𝜑𝐶𝐴)
19 disjf1o.dj . . . . 5 (𝜑Disj 𝑥𝐴 𝐵)
20 disjss1 4783 . . . . 5 (𝐶𝐴 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐶 𝐵))
2118, 19, 20sylc 65 . . . 4 (𝜑Disj 𝑥𝐶 𝐵)
221, 2, 11, 17, 21disjf1 40016 . . 3 (𝜑 → (𝑥𝐶𝐵):𝐶1-1𝑉)
23 f1f1orn 6331 . . 3 ((𝑥𝐶𝐵):𝐶1-1𝑉 → (𝑥𝐶𝐵):𝐶1-1-onto→ran (𝑥𝐶𝐵))
2422, 23syl 17 . 2 (𝜑 → (𝑥𝐶𝐵):𝐶1-1-onto→ran (𝑥𝐶𝐵))
25 disjf1o.f . . . . . 6 𝐹 = (𝑥𝐴𝐵)
2625a1i 11 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐵))
2726reseq1d 5564 . . . 4 (𝜑 → (𝐹𝐶) = ((𝑥𝐴𝐵) ↾ 𝐶))
2818resmptd 5629 . . . 4 (𝜑 → ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥𝐶𝐵))
2927, 28eqtrd 2799 . . 3 (𝜑 → (𝐹𝐶) = (𝑥𝐶𝐵))
30 eqidd 2766 . . 3 (𝜑𝐶 = 𝐶)
31 simpl 474 . . . . . . 7 ((𝜑𝑦𝐷) → 𝜑)
32 id 22 . . . . . . . . . 10 (𝑦𝐷𝑦𝐷)
33 disjf1o.e . . . . . . . . . 10 𝐷 = (ran 𝐹 ∖ {∅})
3432, 33syl6eleq 2854 . . . . . . . . 9 (𝑦𝐷𝑦 ∈ (ran 𝐹 ∖ {∅}))
35 eldifsni 4476 . . . . . . . . 9 (𝑦 ∈ (ran 𝐹 ∖ {∅}) → 𝑦 ≠ ∅)
3634, 35syl 17 . . . . . . . 8 (𝑦𝐷𝑦 ≠ ∅)
3736adantl 473 . . . . . . 7 ((𝜑𝑦𝐷) → 𝑦 ≠ ∅)
38 eldifi 3894 . . . . . . . . . 10 (𝑦 ∈ (ran 𝐹 ∖ {∅}) → 𝑦 ∈ ran 𝐹)
3934, 38syl 17 . . . . . . . . 9 (𝑦𝐷𝑦 ∈ ran 𝐹)
4025elrnmpt 5541 . . . . . . . . . 10 (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝑦 = 𝐵))
4139, 40syl 17 . . . . . . . . 9 (𝑦𝐷 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝑦 = 𝐵))
4239, 41mpbid 223 . . . . . . . 8 (𝑦𝐷 → ∃𝑥𝐴 𝑦 = 𝐵)
4342adantl 473 . . . . . . 7 ((𝜑𝑦𝐷) → ∃𝑥𝐴 𝑦 = 𝐵)
44 nfv 2009 . . . . . . . . . 10 𝑥 𝑦 ≠ ∅
451, 44nfan 1998 . . . . . . . . 9 𝑥(𝜑𝑦 ≠ ∅)
46 nfcv 2907 . . . . . . . . . 10 𝑥𝑦
47 nfmpt1 4906 . . . . . . . . . . 11 𝑥(𝑥𝐶𝐵)
4847nfrn 5537 . . . . . . . . . 10 𝑥ran (𝑥𝐶𝐵)
4946, 48nfel 2920 . . . . . . . . 9 𝑥 𝑦 ∈ ran (𝑥𝐶𝐵)
50 simp3 1168 . . . . . . . . . . . 12 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
51 simp2 1167 . . . . . . . . . . . . . . . 16 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑥𝐴)
52 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵𝑦 = 𝐵)
5352eqcomd 2771 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐵𝐵 = 𝑦)
5453adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝐵 = 𝑦)
55 simpl 474 . . . . . . . . . . . . . . . . . 18 ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
5654, 55eqnetrd 3004 . . . . . . . . . . . . . . . . 17 ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
57563adant2 1161 . . . . . . . . . . . . . . . 16 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝐵 ≠ ∅)
5851, 57jca 507 . . . . . . . . . . . . . . 15 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → (𝑥𝐴𝐵 ≠ ∅))
5958, 13sylibr 225 . . . . . . . . . . . . . 14 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅})
604eqcomi 2774 . . . . . . . . . . . . . . 15 {𝑥𝐴𝐵 ≠ ∅} = 𝐶
6160a1i 11 . . . . . . . . . . . . . 14 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → {𝑥𝐴𝐵 ≠ ∅} = 𝐶)
6259, 61eleqtrd 2846 . . . . . . . . . . . . 13 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑥𝐶)
63 eqvisset 3364 . . . . . . . . . . . . . 14 (𝑦 = 𝐵𝐵 ∈ V)
64633ad2ant3 1165 . . . . . . . . . . . . 13 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝐵 ∈ V)
652elrnmpt1 5543 . . . . . . . . . . . . 13 ((𝑥𝐶𝐵 ∈ V) → 𝐵 ∈ ran (𝑥𝐶𝐵))
6662, 64, 65syl2anc 579 . . . . . . . . . . . 12 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝐵 ∈ ran (𝑥𝐶𝐵))
6750, 66eqeltrd 2844 . . . . . . . . . . 11 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐶𝐵))
68673adant1l 1221 . . . . . . . . . 10 (((𝜑𝑦 ≠ ∅) ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐶𝐵))
69683exp 1148 . . . . . . . . 9 ((𝜑𝑦 ≠ ∅) → (𝑥𝐴 → (𝑦 = 𝐵𝑦 ∈ ran (𝑥𝐶𝐵))))
7045, 49, 69rexlimd 3173 . . . . . . . 8 ((𝜑𝑦 ≠ ∅) → (∃𝑥𝐴 𝑦 = 𝐵𝑦 ∈ ran (𝑥𝐶𝐵)))
7170imp 395 . . . . . . 7 (((𝜑𝑦 ≠ ∅) ∧ ∃𝑥𝐴 𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐶𝐵))
7231, 37, 43, 71syl21anc 866 . . . . . 6 ((𝜑𝑦𝐷) → 𝑦 ∈ ran (𝑥𝐶𝐵))
7372ralrimiva 3113 . . . . 5 (𝜑 → ∀𝑦𝐷 𝑦 ∈ ran (𝑥𝐶𝐵))
74 dfss3 3750 . . . . 5 (𝐷 ⊆ ran (𝑥𝐶𝐵) ↔ ∀𝑦𝐷 𝑦 ∈ ran (𝑥𝐶𝐵))
7573, 74sylibr 225 . . . 4 (𝜑𝐷 ⊆ ran (𝑥𝐶𝐵))
76 simpl 474 . . . . 5 ((𝜑𝑦 ∈ ran (𝑥𝐶𝐵)) → 𝜑)
77 vex 3353 . . . . . . . 8 𝑦 ∈ V
782elrnmpt 5541 . . . . . . . 8 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝐶𝐵) ↔ ∃𝑥𝐶 𝑦 = 𝐵))
7977, 78ax-mp 5 . . . . . . 7 (𝑦 ∈ ran (𝑥𝐶𝐵) ↔ ∃𝑥𝐶 𝑦 = 𝐵)
8079biimpi 207 . . . . . 6 (𝑦 ∈ ran (𝑥𝐶𝐵) → ∃𝑥𝐶 𝑦 = 𝐵)
8180adantl 473 . . . . 5 ((𝜑𝑦 ∈ ran (𝑥𝐶𝐵)) → ∃𝑥𝐶 𝑦 = 𝐵)
82 nfv 2009 . . . . . . 7 𝑥 𝑦𝐷
83 simpr 477 . . . . . . . . . . . 12 ((𝑥𝐶𝑦 = 𝐵) → 𝑦 = 𝐵)
848adantr 472 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦 = 𝐵) → 𝑥𝐴)
8583, 63syl 17 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦 = 𝐵) → 𝐵 ∈ V)
8625elrnmpt1 5543 . . . . . . . . . . . . 13 ((𝑥𝐴𝐵 ∈ V) → 𝐵 ∈ ran 𝐹)
8784, 85, 86syl2anc 579 . . . . . . . . . . . 12 ((𝑥𝐶𝑦 = 𝐵) → 𝐵 ∈ ran 𝐹)
8883, 87eqeltrd 2844 . . . . . . . . . . 11 ((𝑥𝐶𝑦 = 𝐵) → 𝑦 ∈ ran 𝐹)
89883adant1 1160 . . . . . . . . . 10 ((𝜑𝑥𝐶𝑦 = 𝐵) → 𝑦 ∈ ran 𝐹)
9016adantr 472 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦 = 𝐵) → 𝐵 ≠ ∅)
9183, 90eqnetrd 3004 . . . . . . . . . . . 12 ((𝑥𝐶𝑦 = 𝐵) → 𝑦 ≠ ∅)
92 nelsn 4370 . . . . . . . . . . . 12 (𝑦 ≠ ∅ → ¬ 𝑦 ∈ {∅})
9391, 92syl 17 . . . . . . . . . . 11 ((𝑥𝐶𝑦 = 𝐵) → ¬ 𝑦 ∈ {∅})
94933adant1 1160 . . . . . . . . . 10 ((𝜑𝑥𝐶𝑦 = 𝐵) → ¬ 𝑦 ∈ {∅})
9589, 94eldifd 3743 . . . . . . . . 9 ((𝜑𝑥𝐶𝑦 = 𝐵) → 𝑦 ∈ (ran 𝐹 ∖ {∅}))
9695, 33syl6eleqr 2855 . . . . . . . 8 ((𝜑𝑥𝐶𝑦 = 𝐵) → 𝑦𝐷)
97963exp 1148 . . . . . . 7 (𝜑 → (𝑥𝐶 → (𝑦 = 𝐵𝑦𝐷)))
981, 82, 97rexlimd 3173 . . . . . 6 (𝜑 → (∃𝑥𝐶 𝑦 = 𝐵𝑦𝐷))
9998imp 395 . . . . 5 ((𝜑 ∧ ∃𝑥𝐶 𝑦 = 𝐵) → 𝑦𝐷)
10076, 81, 99syl2anc 579 . . . 4 ((𝜑𝑦 ∈ ran (𝑥𝐶𝐵)) → 𝑦𝐷)
10175, 100eqelssd 3782 . . 3 (𝜑𝐷 = ran (𝑥𝐶𝐵))
10229, 30, 101f1oeq123d 6316 . 2 (𝜑 → ((𝐹𝐶):𝐶1-1-onto𝐷 ↔ (𝑥𝐶𝐵):𝐶1-1-onto→ran (𝑥𝐶𝐵)))
10324, 102mpbird 248 1 (𝜑 → (𝐹𝐶):𝐶1-1-onto𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wnf 1878  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3729  wss 3732  c0 4079  {csn 4334  Disj wdisj 4777  cmpt 4888  ran crn 5278  cres 5279  1-1wf1 6065  1-1-ontowf1o 6067
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-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
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-rmo 3063  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-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-disj 4778  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
This theorem is referenced by:  sge0fodjrnlem  41270
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