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Theorem disjf1o 39692
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 2651 . . . 4 (𝑥𝐶𝐵) = (𝑥𝐶𝐵)
3 simpl 472 . . . . 5 ((𝜑𝑥𝐶) → 𝜑)
4 disjf1o.d . . . . . . . 8 𝐶 = {𝑥𝐴𝐵 ≠ ∅}
5 ssrab2 3720 . . . . . . . 8 {𝑥𝐴𝐵 ≠ ∅} ⊆ 𝐴
64, 5eqsstri 3668 . . . . . . 7 𝐶𝐴
7 id 22 . . . . . . 7 (𝑥𝐶𝑥𝐶)
86, 7sseldi 3634 . . . . . 6 (𝑥𝐶𝑥𝐴)
98adantl 481 . . . . 5 ((𝜑𝑥𝐶) → 𝑥𝐴)
10 disjf1o.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
113, 9, 10syl2anc 694 . . . 4 ((𝜑𝑥𝐶) → 𝐵𝑉)
127, 4syl6eleq 2740 . . . . . . 7 (𝑥𝐶𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅})
13 rabid 3145 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅} ↔ (𝑥𝐴𝐵 ≠ ∅))
1413a1i 11 . . . . . . 7 (𝑥𝐶 → (𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅} ↔ (𝑥𝐴𝐵 ≠ ∅)))
1512, 14mpbid 222 . . . . . 6 (𝑥𝐶 → (𝑥𝐴𝐵 ≠ ∅))
1615simprd 478 . . . . 5 (𝑥𝐶𝐵 ≠ ∅)
1716adantl 481 . . . 4 ((𝜑𝑥𝐶) → 𝐵 ≠ ∅)
186a1i 11 . . . . 5 (𝜑𝐶𝐴)
19 disjf1o.dj . . . . 5 (𝜑Disj 𝑥𝐴 𝐵)
20 disjss1 4658 . . . . 5 (𝐶𝐴 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐶 𝐵))
2118, 19, 20sylc 65 . . . 4 (𝜑Disj 𝑥𝐶 𝐵)
221, 2, 11, 17, 21disjf1 39683 . . 3 (𝜑 → (𝑥𝐶𝐵):𝐶1-1𝑉)
23 f1f1orn 6186 . . 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 5427 . . . 4 (𝜑 → (𝐹𝐶) = ((𝑥𝐴𝐵) ↾ 𝐶))
2818resmptd 5487 . . . 4 (𝜑 → ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥𝐶𝐵))
2927, 28eqtrd 2685 . . 3 (𝜑 → (𝐹𝐶) = (𝑥𝐶𝐵))
30 eqidd 2652 . . 3 (𝜑𝐶 = 𝐶)
31 simpl 472 . . . . . . 7 ((𝜑𝑦𝐷) → 𝜑)
32 id 22 . . . . . . . . . 10 (𝑦𝐷𝑦𝐷)
33 disjf1o.e . . . . . . . . . 10 𝐷 = (ran 𝐹 ∖ {∅})
3432, 33syl6eleq 2740 . . . . . . . . 9 (𝑦𝐷𝑦 ∈ (ran 𝐹 ∖ {∅}))
35 eldifsni 4353 . . . . . . . . 9 (𝑦 ∈ (ran 𝐹 ∖ {∅}) → 𝑦 ≠ ∅)
3634, 35syl 17 . . . . . . . 8 (𝑦𝐷𝑦 ≠ ∅)
3736adantl 481 . . . . . . 7 ((𝜑𝑦𝐷) → 𝑦 ≠ ∅)
38 eldifi 3765 . . . . . . . . . 10 (𝑦 ∈ (ran 𝐹 ∖ {∅}) → 𝑦 ∈ ran 𝐹)
3934, 38syl 17 . . . . . . . . 9 (𝑦𝐷𝑦 ∈ ran 𝐹)
4025elrnmpt 5404 . . . . . . . . . 10 (𝑦 ∈ ran 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝑦 = 𝐵))
4139, 40syl 17 . . . . . . . . 9 (𝑦𝐷 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝑦 = 𝐵))
4239, 41mpbid 222 . . . . . . . 8 (𝑦𝐷 → ∃𝑥𝐴 𝑦 = 𝐵)
4342adantl 481 . . . . . . 7 ((𝜑𝑦𝐷) → ∃𝑥𝐴 𝑦 = 𝐵)
44 nfv 1883 . . . . . . . . . 10 𝑥 𝑦 ≠ ∅
451, 44nfan 1868 . . . . . . . . 9 𝑥(𝜑𝑦 ≠ ∅)
46 nfcv 2793 . . . . . . . . . 10 𝑥𝑦
47 nfmpt1 4780 . . . . . . . . . . 11 𝑥(𝑥𝐶𝐵)
4847nfrn 5400 . . . . . . . . . 10 𝑥ran (𝑥𝐶𝐵)
4946, 48nfel 2806 . . . . . . . . 9 𝑥 𝑦 ∈ ran (𝑥𝐶𝐵)
50 simp3 1083 . . . . . . . . . . . 12 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
51 simp2 1082 . . . . . . . . . . . . . . . 16 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑥𝐴)
52 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵𝑦 = 𝐵)
5352eqcomd 2657 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐵𝐵 = 𝑦)
5453adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝐵 = 𝑦)
55 simpl 472 . . . . . . . . . . . . . . . . . 18 ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
5654, 55eqnetrd 2890 . . . . . . . . . . . . . . . . 17 ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
57563adant2 1100 . . . . . . . . . . . . . . . 16 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝐵 ≠ ∅)
5851, 57jca 553 . . . . . . . . . . . . . . 15 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → (𝑥𝐴𝐵 ≠ ∅))
5958, 13sylibr 224 . . . . . . . . . . . . . 14 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅})
604eqcomi 2660 . . . . . . . . . . . . . . 15 {𝑥𝐴𝐵 ≠ ∅} = 𝐶
6160a1i 11 . . . . . . . . . . . . . 14 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → {𝑥𝐴𝐵 ≠ ∅} = 𝐶)
6259, 61eleqtrd 2732 . . . . . . . . . . . . 13 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑥𝐶)
63 eqvisset 3242 . . . . . . . . . . . . . 14 (𝑦 = 𝐵𝐵 ∈ V)
64633ad2ant3 1104 . . . . . . . . . . . . 13 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝐵 ∈ V)
652elrnmpt1 5406 . . . . . . . . . . . . 13 ((𝑥𝐶𝐵 ∈ V) → 𝐵 ∈ ran (𝑥𝐶𝐵))
6662, 64, 65syl2anc 694 . . . . . . . . . . . 12 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝐵 ∈ ran (𝑥𝐶𝐵))
6750, 66eqeltrd 2730 . . . . . . . . . . 11 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐶𝐵))
68673adant1l 1358 . . . . . . . . . 10 (((𝜑𝑦 ≠ ∅) ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐶𝐵))
69683exp 1283 . . . . . . . . 9 ((𝜑𝑦 ≠ ∅) → (𝑥𝐴 → (𝑦 = 𝐵𝑦 ∈ ran (𝑥𝐶𝐵))))
7045, 49, 69rexlimd 3055 . . . . . . . 8 ((𝜑𝑦 ≠ ∅) → (∃𝑥𝐴 𝑦 = 𝐵𝑦 ∈ ran (𝑥𝐶𝐵)))
7170imp 444 . . . . . . 7 (((𝜑𝑦 ≠ ∅) ∧ ∃𝑥𝐴 𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐶𝐵))
7231, 37, 43, 71syl21anc 1365 . . . . . 6 ((𝜑𝑦𝐷) → 𝑦 ∈ ran (𝑥𝐶𝐵))
7372ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑦𝐷 𝑦 ∈ ran (𝑥𝐶𝐵))
74 dfss3 3625 . . . . 5 (𝐷 ⊆ ran (𝑥𝐶𝐵) ↔ ∀𝑦𝐷 𝑦 ∈ ran (𝑥𝐶𝐵))
7573, 74sylibr 224 . . . 4 (𝜑𝐷 ⊆ ran (𝑥𝐶𝐵))
76 simpl 472 . . . . . . 7 ((𝜑𝑦 ∈ ran (𝑥𝐶𝐵)) → 𝜑)
77 vex 3234 . . . . . . . . . 10 𝑦 ∈ V
782elrnmpt 5404 . . . . . . . . . 10 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝐶𝐵) ↔ ∃𝑥𝐶 𝑦 = 𝐵))
7977, 78ax-mp 5 . . . . . . . . 9 (𝑦 ∈ ran (𝑥𝐶𝐵) ↔ ∃𝑥𝐶 𝑦 = 𝐵)
8079biimpi 206 . . . . . . . 8 (𝑦 ∈ ran (𝑥𝐶𝐵) → ∃𝑥𝐶 𝑦 = 𝐵)
8180adantl 481 . . . . . . 7 ((𝜑𝑦 ∈ ran (𝑥𝐶𝐵)) → ∃𝑥𝐶 𝑦 = 𝐵)
82 nfv 1883 . . . . . . . . 9 𝑥 𝑦𝐷
83 simpr 476 . . . . . . . . . . . . . 14 ((𝑥𝐶𝑦 = 𝐵) → 𝑦 = 𝐵)
848adantr 480 . . . . . . . . . . . . . . 15 ((𝑥𝐶𝑦 = 𝐵) → 𝑥𝐴)
8583, 63syl 17 . . . . . . . . . . . . . . 15 ((𝑥𝐶𝑦 = 𝐵) → 𝐵 ∈ V)
8625elrnmpt1 5406 . . . . . . . . . . . . . . 15 ((𝑥𝐴𝐵 ∈ V) → 𝐵 ∈ ran 𝐹)
8784, 85, 86syl2anc 694 . . . . . . . . . . . . . 14 ((𝑥𝐶𝑦 = 𝐵) → 𝐵 ∈ ran 𝐹)
8883, 87eqeltrd 2730 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦 = 𝐵) → 𝑦 ∈ ran 𝐹)
89883adant1 1099 . . . . . . . . . . . 12 ((𝜑𝑥𝐶𝑦 = 𝐵) → 𝑦 ∈ ran 𝐹)
9016adantr 480 . . . . . . . . . . . . . . 15 ((𝑥𝐶𝑦 = 𝐵) → 𝐵 ≠ ∅)
9183, 90eqnetrd 2890 . . . . . . . . . . . . . 14 ((𝑥𝐶𝑦 = 𝐵) → 𝑦 ≠ ∅)
92 nelsn 4245 . . . . . . . . . . . . . 14 (𝑦 ≠ ∅ → ¬ 𝑦 ∈ {∅})
9391, 92syl 17 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦 = 𝐵) → ¬ 𝑦 ∈ {∅})
94933adant1 1099 . . . . . . . . . . . 12 ((𝜑𝑥𝐶𝑦 = 𝐵) → ¬ 𝑦 ∈ {∅})
9589, 94eldifd 3618 . . . . . . . . . . 11 ((𝜑𝑥𝐶𝑦 = 𝐵) → 𝑦 ∈ (ran 𝐹 ∖ {∅}))
9695, 33syl6eleqr 2741 . . . . . . . . . 10 ((𝜑𝑥𝐶𝑦 = 𝐵) → 𝑦𝐷)
97963exp 1283 . . . . . . . . 9 (𝜑 → (𝑥𝐶 → (𝑦 = 𝐵𝑦𝐷)))
981, 82, 97rexlimd 3055 . . . . . . . 8 (𝜑 → (∃𝑥𝐶 𝑦 = 𝐵𝑦𝐷))
9998imp 444 . . . . . . 7 ((𝜑 ∧ ∃𝑥𝐶 𝑦 = 𝐵) → 𝑦𝐷)
10076, 81, 99syl2anc 694 . . . . . 6 ((𝜑𝑦 ∈ ran (𝑥𝐶𝐵)) → 𝑦𝐷)
101100ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑦 ∈ ran (𝑥𝐶𝐵)𝑦𝐷)
102 dfss3 3625 . . . . 5 (ran (𝑥𝐶𝐵) ⊆ 𝐷 ↔ ∀𝑦 ∈ ran (𝑥𝐶𝐵)𝑦𝐷)
103101, 102sylibr 224 . . . 4 (𝜑 → ran (𝑥𝐶𝐵) ⊆ 𝐷)
10475, 103eqssd 3653 . . 3 (𝜑𝐷 = ran (𝑥𝐶𝐵))
10529, 30, 104f1oeq123d 6171 . 2 (𝜑 → ((𝐹𝐶):𝐶1-1-onto𝐷 ↔ (𝑥𝐶𝐵):𝐶1-1-onto→ran (𝑥𝐶𝐵)))
10624, 105mpbird 247 1 (𝜑 → (𝐹𝐶):𝐶1-1-onto𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wnf 1748  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  c0 3948  {csn 4210  Disj wdisj 4652  cmpt 4762  ran crn 5144  cres 5145  1-1wf1 5923  1-1-ontowf1o 5925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  sge0fodjrnlem  40951
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