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Theorem disjf1o 38838
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 2626 . . . 4 (𝑥𝐶𝐵) = (𝑥𝐶𝐵)
3 simpl 473 . . . . 5 ((𝜑𝑥𝐶) → 𝜑)
4 disjf1o.d . . . . . . . 8 𝐶 = {𝑥𝐴𝐵 ≠ ∅}
5 ssrab2 3671 . . . . . . . 8 {𝑥𝐴𝐵 ≠ ∅} ⊆ 𝐴
64, 5eqsstri 3619 . . . . . . 7 𝐶𝐴
7 id 22 . . . . . . 7 (𝑥𝐶𝑥𝐶)
86, 7sseldi 3586 . . . . . 6 (𝑥𝐶𝑥𝐴)
98adantl 482 . . . . 5 ((𝜑𝑥𝐶) → 𝑥𝐴)
10 disjf1o.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
113, 9, 10syl2anc 692 . . . 4 ((𝜑𝑥𝐶) → 𝐵𝑉)
127, 4syl6eleq 2714 . . . . . . 7 (𝑥𝐶𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅})
13 rabid 3111 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅} ↔ (𝑥𝐴𝐵 ≠ ∅))
1413a1i 11 . . . . . . 7 (𝑥𝐶 → (𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅} ↔ (𝑥𝐴𝐵 ≠ ∅)))
1512, 14mpbid 222 . . . . . 6 (𝑥𝐶 → (𝑥𝐴𝐵 ≠ ∅))
1615simprd 479 . . . . 5 (𝑥𝐶𝐵 ≠ ∅)
1716adantl 482 . . . 4 ((𝜑𝑥𝐶) → 𝐵 ≠ ∅)
186a1i 11 . . . . 5 (𝜑𝐶𝐴)
19 disjf1o.dj . . . . 5 (𝜑Disj 𝑥𝐴 𝐵)
20 disjss1 4594 . . . . 5 (𝐶𝐴 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐶 𝐵))
2118, 19, 20sylc 65 . . . 4 (𝜑Disj 𝑥𝐶 𝐵)
221, 2, 11, 17, 21disjf1 38829 . . 3 (𝜑 → (𝑥𝐶𝐵):𝐶1-1𝑉)
23 f1f1orn 6107 . . 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 5359 . . . 4 (𝜑 → (𝐹𝐶) = ((𝑥𝐴𝐵) ↾ 𝐶))
2818resmptd 5415 . . . 4 (𝜑 → ((𝑥𝐴𝐵) ↾ 𝐶) = (𝑥𝐶𝐵))
2927, 28eqtrd 2660 . . 3 (𝜑 → (𝐹𝐶) = (𝑥𝐶𝐵))
30 eqidd 2627 . . 3 (𝜑𝐶 = 𝐶)
31 simpl 473 . . . . . . 7 ((𝜑𝑦𝐷) → 𝜑)
32 id 22 . . . . . . . . . 10 (𝑦𝐷𝑦𝐷)
33 disjf1o.e . . . . . . . . . 10 𝐷 = (ran 𝐹 ∖ {∅})
3432, 33syl6eleq 2714 . . . . . . . . 9 (𝑦𝐷𝑦 ∈ (ran 𝐹 ∖ {∅}))
35 eldifsni 4294 . . . . . . . . 9 (𝑦 ∈ (ran 𝐹 ∖ {∅}) → 𝑦 ≠ ∅)
3634, 35syl 17 . . . . . . . 8 (𝑦𝐷𝑦 ≠ ∅)
3736adantl 482 . . . . . . 7 ((𝜑𝑦𝐷) → 𝑦 ≠ ∅)
38 eldifi 3715 . . . . . . . . . 10 (𝑦 ∈ (ran 𝐹 ∖ {∅}) → 𝑦 ∈ ran 𝐹)
3934, 38syl 17 . . . . . . . . 9 (𝑦𝐷𝑦 ∈ ran 𝐹)
4025elrnmpt 5336 . . . . . . . . . 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 482 . . . . . . 7 ((𝜑𝑦𝐷) → ∃𝑥𝐴 𝑦 = 𝐵)
44 nfv 1845 . . . . . . . . . 10 𝑥 𝑦 ≠ ∅
451, 44nfan 1830 . . . . . . . . 9 𝑥(𝜑𝑦 ≠ ∅)
46 nfcv 2767 . . . . . . . . . 10 𝑥𝑦
47 nfmpt1 4712 . . . . . . . . . . 11 𝑥(𝑥𝐶𝐵)
4847nfrn 5332 . . . . . . . . . 10 𝑥ran (𝑥𝐶𝐵)
4946, 48nfel 2779 . . . . . . . . 9 𝑥 𝑦 ∈ ran (𝑥𝐶𝐵)
50 simp3 1061 . . . . . . . . . . . 12 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
51 simp2 1060 . . . . . . . . . . . . . . . 16 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑥𝐴)
52 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐵𝑦 = 𝐵)
5352eqcomd 2632 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐵𝐵 = 𝑦)
5453adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝐵 = 𝑦)
55 simpl 473 . . . . . . . . . . . . . . . . . 18 ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
5654, 55eqnetrd 2863 . . . . . . . . . . . . . . . . 17 ((𝑦 ≠ ∅ ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
57563adant2 1078 . . . . . . . . . . . . . . . 16 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝐵 ≠ ∅)
5851, 57jca 554 . . . . . . . . . . . . . . 15 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → (𝑥𝐴𝐵 ≠ ∅))
5958, 13sylibr 224 . . . . . . . . . . . . . 14 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑥 ∈ {𝑥𝐴𝐵 ≠ ∅})
604eqcomi 2635 . . . . . . . . . . . . . . 15 {𝑥𝐴𝐵 ≠ ∅} = 𝐶
6160a1i 11 . . . . . . . . . . . . . 14 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → {𝑥𝐴𝐵 ≠ ∅} = 𝐶)
6259, 61eleqtrd 2706 . . . . . . . . . . . . 13 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑥𝐶)
63 eqvisset 3202 . . . . . . . . . . . . . 14 (𝑦 = 𝐵𝐵 ∈ V)
64633ad2ant3 1082 . . . . . . . . . . . . 13 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝐵 ∈ V)
652elrnmpt1 5338 . . . . . . . . . . . . 13 ((𝑥𝐶𝐵 ∈ V) → 𝐵 ∈ ran (𝑥𝐶𝐵))
6662, 64, 65syl2anc 692 . . . . . . . . . . . 12 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝐵 ∈ ran (𝑥𝐶𝐵))
6750, 66eqeltrd 2704 . . . . . . . . . . 11 ((𝑦 ≠ ∅ ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐶𝐵))
68673adant1l 1315 . . . . . . . . . 10 (((𝜑𝑦 ≠ ∅) ∧ 𝑥𝐴𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐶𝐵))
69683exp 1261 . . . . . . . . 9 ((𝜑𝑦 ≠ ∅) → (𝑥𝐴 → (𝑦 = 𝐵𝑦 ∈ ran (𝑥𝐶𝐵))))
7045, 49, 69rexlimd 3024 . . . . . . . 8 ((𝜑𝑦 ≠ ∅) → (∃𝑥𝐴 𝑦 = 𝐵𝑦 ∈ ran (𝑥𝐶𝐵)))
7170imp 445 . . . . . . 7 (((𝜑𝑦 ≠ ∅) ∧ ∃𝑥𝐴 𝑦 = 𝐵) → 𝑦 ∈ ran (𝑥𝐶𝐵))
7231, 37, 43, 71syl21anc 1322 . . . . . 6 ((𝜑𝑦𝐷) → 𝑦 ∈ ran (𝑥𝐶𝐵))
7372ralrimiva 2965 . . . . 5 (𝜑 → ∀𝑦𝐷 𝑦 ∈ ran (𝑥𝐶𝐵))
74 dfss3 3578 . . . . 5 (𝐷 ⊆ ran (𝑥𝐶𝐵) ↔ ∀𝑦𝐷 𝑦 ∈ ran (𝑥𝐶𝐵))
7573, 74sylibr 224 . . . 4 (𝜑𝐷 ⊆ ran (𝑥𝐶𝐵))
76 simpl 473 . . . . . . 7 ((𝜑𝑦 ∈ ran (𝑥𝐶𝐵)) → 𝜑)
77 vex 3194 . . . . . . . . . 10 𝑦 ∈ V
782elrnmpt 5336 . . . . . . . . . 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 482 . . . . . . 7 ((𝜑𝑦 ∈ ran (𝑥𝐶𝐵)) → ∃𝑥𝐶 𝑦 = 𝐵)
82 nfv 1845 . . . . . . . . 9 𝑥 𝑦𝐷
83 simpr 477 . . . . . . . . . . . . . 14 ((𝑥𝐶𝑦 = 𝐵) → 𝑦 = 𝐵)
848adantr 481 . . . . . . . . . . . . . . 15 ((𝑥𝐶𝑦 = 𝐵) → 𝑥𝐴)
8583, 63syl 17 . . . . . . . . . . . . . . 15 ((𝑥𝐶𝑦 = 𝐵) → 𝐵 ∈ V)
8625elrnmpt1 5338 . . . . . . . . . . . . . . 15 ((𝑥𝐴𝐵 ∈ V) → 𝐵 ∈ ran 𝐹)
8784, 85, 86syl2anc 692 . . . . . . . . . . . . . 14 ((𝑥𝐶𝑦 = 𝐵) → 𝐵 ∈ ran 𝐹)
8883, 87eqeltrd 2704 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦 = 𝐵) → 𝑦 ∈ ran 𝐹)
89883adant1 1077 . . . . . . . . . . . 12 ((𝜑𝑥𝐶𝑦 = 𝐵) → 𝑦 ∈ ran 𝐹)
9016adantr 481 . . . . . . . . . . . . . . 15 ((𝑥𝐶𝑦 = 𝐵) → 𝐵 ≠ ∅)
9183, 90eqnetrd 2863 . . . . . . . . . . . . . 14 ((𝑥𝐶𝑦 = 𝐵) → 𝑦 ≠ ∅)
92 nelsn 4188 . . . . . . . . . . . . . 14 (𝑦 ≠ ∅ → ¬ 𝑦 ∈ {∅})
9391, 92syl 17 . . . . . . . . . . . . 13 ((𝑥𝐶𝑦 = 𝐵) → ¬ 𝑦 ∈ {∅})
94933adant1 1077 . . . . . . . . . . . 12 ((𝜑𝑥𝐶𝑦 = 𝐵) → ¬ 𝑦 ∈ {∅})
9589, 94eldifd 3571 . . . . . . . . . . 11 ((𝜑𝑥𝐶𝑦 = 𝐵) → 𝑦 ∈ (ran 𝐹 ∖ {∅}))
9695, 33syl6eleqr 2715 . . . . . . . . . 10 ((𝜑𝑥𝐶𝑦 = 𝐵) → 𝑦𝐷)
97963exp 1261 . . . . . . . . 9 (𝜑 → (𝑥𝐶 → (𝑦 = 𝐵𝑦𝐷)))
981, 82, 97rexlimd 3024 . . . . . . . 8 (𝜑 → (∃𝑥𝐶 𝑦 = 𝐵𝑦𝐷))
9998imp 445 . . . . . . 7 ((𝜑 ∧ ∃𝑥𝐶 𝑦 = 𝐵) → 𝑦𝐷)
10076, 81, 99syl2anc 692 . . . . . 6 ((𝜑𝑦 ∈ ran (𝑥𝐶𝐵)) → 𝑦𝐷)
101100ralrimiva 2965 . . . . 5 (𝜑 → ∀𝑦 ∈ ran (𝑥𝐶𝐵)𝑦𝐷)
102 dfss3 3578 . . . . 5 (ran (𝑥𝐶𝐵) ⊆ 𝐷 ↔ ∀𝑦 ∈ ran (𝑥𝐶𝐵)𝑦𝐷)
103101, 102sylibr 224 . . . 4 (𝜑 → ran (𝑥𝐶𝐵) ⊆ 𝐷)
10475, 103eqssd 3605 . . 3 (𝜑𝐷 = ran (𝑥𝐶𝐵))
10529, 30, 104f1oeq123d 6092 . 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 384  w3a 1036   = wceq 1480  wnf 1705  wcel 1992  wne 2796  wral 2912  wrex 2913  {crab 2916  Vcvv 3191  cdif 3557  wss 3560  c0 3896  {csn 4153  Disj wdisj 4588  cmpt 4678  ran crn 5080  cres 5081  1-1wf1 5847  1-1-ontowf1o 5849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858
This theorem is referenced by:  sge0fodjrnlem  39927
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