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Theorem disjf1 43928
Description: A 1 to 1 mapping built from disjoint, nonempty sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
disjf1.xph 𝑥𝜑
disjf1.f 𝐹 = (𝑥𝐴𝐵)
disjf1.b ((𝜑𝑥𝐴) → 𝐵𝑉)
disjf1.n0 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
disjf1.dj (𝜑Disj 𝑥𝐴 𝐵)
Assertion
Ref Expression
disjf1 (𝜑𝐹:𝐴1-1𝑉)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem disjf1
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 disjf1.xph . . . . . . 7 𝑥𝜑
2 nfv 1918 . . . . . . 7 𝑥 𝑦𝐴
31, 2nfan 1903 . . . . . 6 𝑥(𝜑𝑦𝐴)
4 nfcsb1v 3919 . . . . . . 7 𝑥𝑦 / 𝑥𝐵
5 nfcv 2904 . . . . . . 7 𝑥𝑉
64, 5nfel 2918 . . . . . 6 𝑥𝑦 / 𝑥𝐵𝑉
73, 6nfim 1900 . . . . 5 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)
8 eleq1w 2817 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98anbi2d 630 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
10 csbeq1a 3908 . . . . . . 7 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1110eleq1d 2819 . . . . . 6 (𝑥 = 𝑦 → (𝐵𝑉𝑦 / 𝑥𝐵𝑉))
129, 11imbi12d 345 . . . . 5 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝐵𝑉) ↔ ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)))
13 disjf1.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
147, 12, 13chvarfv 2234 . . . 4 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)
1514ralrimiva 3147 . . 3 (𝜑 → ∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉)
16 inidm 4219 . . . . . . . . 9 (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = 𝑦 / 𝑥𝐵
1716eqcomi 2742 . . . . . . . 8 𝑦 / 𝑥𝐵 = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵)
1817a1i 11 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵))
19 ineq2 4207 . . . . . . . 8 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵))
2019ad2antlr 726 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵))
21 disjf1.dj . . . . . . . . . 10 (𝜑Disj 𝑥𝐴 𝐵)
22 nfcv 2904 . . . . . . . . . . 11 𝑤𝐵
23 nfcsb1v 3919 . . . . . . . . . . 11 𝑥𝑤 / 𝑥𝐵
24 csbeq1a 3908 . . . . . . . . . . 11 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
2522, 23, 24cbvdisj 5124 . . . . . . . . . 10 (Disj 𝑥𝐴 𝐵Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
2621, 25sylib 217 . . . . . . . . 9 (𝜑Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
2726ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
28 simpllr 775 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦𝐴𝑧𝐴))
29 neqne 2949 . . . . . . . . 9 𝑦 = 𝑧𝑦𝑧)
3029adantl 483 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦𝑧)
31 csbeq1 3897 . . . . . . . . 9 (𝑤 = 𝑦𝑤 / 𝑥𝐵 = 𝑦 / 𝑥𝐵)
32 csbeq1 3897 . . . . . . . . 9 (𝑤 = 𝑧𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3331, 32disji2 5131 . . . . . . . 8 ((Disj 𝑤𝐴 𝑤 / 𝑥𝐵 ∧ (𝑦𝐴𝑧𝐴) ∧ 𝑦𝑧) → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅)
3427, 28, 30, 33syl3anc 1372 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅)
3518, 20, 343eqtrd 2777 . . . . . 6 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 = ∅)
36 nfcv 2904 . . . . . . . . . . . 12 𝑥
374, 36nfne 3044 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵 ≠ ∅
383, 37nfim 1900 . . . . . . . . . 10 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)
3910neeq1d 3001 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐵 ≠ ∅ ↔ 𝑦 / 𝑥𝐵 ≠ ∅))
409, 39imbi12d 345 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝐵 ≠ ∅) ↔ ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)))
41 disjf1.n0 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
4238, 40, 41chvarfv 2234 . . . . . . . . 9 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)
4342adantrr 716 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → 𝑦 / 𝑥𝐵 ≠ ∅)
4443ad2antrr 725 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 ≠ ∅)
4544neneqd 2946 . . . . . 6 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → ¬ 𝑦 / 𝑥𝐵 = ∅)
4635, 45condan 817 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) → 𝑦 = 𝑧)
4746ex 414 . . . 4 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧))
4847ralrimivva 3201 . . 3 (𝜑 → ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧))
4915, 48jca 513 . 2 (𝜑 → (∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧)))
50 disjf1.f . . . 4 𝐹 = (𝑥𝐴𝐵)
51 nfcv 2904 . . . . 5 𝑦𝐵
5251, 4, 10cbvmpt 5260 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
5350, 52eqtri 2761 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
54 csbeq1 3897 . . 3 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
5553, 54f1mpt 7260 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧)))
5649, 55sylibr 233 1 (𝜑𝐹:𝐴1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wnf 1786  wcel 2107  wne 2941  wral 3062  csb 3894  cin 3948  c0 4323  Disj wdisj 5114  cmpt 5232  1-1wf1 6541
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 5300  ax-nul 5307  ax-pr 5428
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fv 6552
This theorem is referenced by:  disjf1o  43937  meadjiunlem  45229
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