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Theorem disjf1 45188
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 1914 . . . . . . 7 𝑥 𝑦𝐴
31, 2nfan 1899 . . . . . 6 𝑥(𝜑𝑦𝐴)
4 nfcsb1v 3923 . . . . . . 7 𝑥𝑦 / 𝑥𝐵
5 nfcv 2905 . . . . . . 7 𝑥𝑉
64, 5nfel 2920 . . . . . 6 𝑥𝑦 / 𝑥𝐵𝑉
73, 6nfim 1896 . . . . 5 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)
8 eleq1w 2824 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98anbi2d 630 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
10 csbeq1a 3913 . . . . . . 7 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1110eleq1d 2826 . . . . . 6 (𝑥 = 𝑦 → (𝐵𝑉𝑦 / 𝑥𝐵𝑉))
129, 11imbi12d 344 . . . . 5 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝐵𝑉) ↔ ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)))
13 disjf1.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
147, 12, 13chvarfv 2240 . . . 4 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)
1514ralrimiva 3146 . . 3 (𝜑 → ∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉)
16 inidm 4227 . . . . . . . . 9 (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = 𝑦 / 𝑥𝐵
1716eqcomi 2746 . . . . . . . 8 𝑦 / 𝑥𝐵 = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵)
1817a1i 11 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵))
19 ineq2 4214 . . . . . . . 8 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵))
2019ad2antlr 727 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵))
21 disjf1.dj . . . . . . . . . 10 (𝜑Disj 𝑥𝐴 𝐵)
22 nfcv 2905 . . . . . . . . . . 11 𝑤𝐵
23 nfcsb1v 3923 . . . . . . . . . . 11 𝑥𝑤 / 𝑥𝐵
24 csbeq1a 3913 . . . . . . . . . . 11 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
2522, 23, 24cbvdisj 5120 . . . . . . . . . 10 (Disj 𝑥𝐴 𝐵Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
2621, 25sylib 218 . . . . . . . . 9 (𝜑Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
2726ad3antrrr 730 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
28 simpllr 776 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦𝐴𝑧𝐴))
29 neqne 2948 . . . . . . . . 9 𝑦 = 𝑧𝑦𝑧)
3029adantl 481 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦𝑧)
31 csbeq1 3902 . . . . . . . . 9 (𝑤 = 𝑦𝑤 / 𝑥𝐵 = 𝑦 / 𝑥𝐵)
32 csbeq1 3902 . . . . . . . . 9 (𝑤 = 𝑧𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3331, 32disji2 5127 . . . . . . . 8 ((Disj 𝑤𝐴 𝑤 / 𝑥𝐵 ∧ (𝑦𝐴𝑧𝐴) ∧ 𝑦𝑧) → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅)
3427, 28, 30, 33syl3anc 1373 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅)
3518, 20, 343eqtrd 2781 . . . . . 6 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 = ∅)
36 nfcv 2905 . . . . . . . . . . . 12 𝑥
374, 36nfne 3043 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵 ≠ ∅
383, 37nfim 1896 . . . . . . . . . 10 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)
3910neeq1d 3000 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐵 ≠ ∅ ↔ 𝑦 / 𝑥𝐵 ≠ ∅))
409, 39imbi12d 344 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝐵 ≠ ∅) ↔ ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)))
41 disjf1.n0 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
4238, 40, 41chvarfv 2240 . . . . . . . . 9 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)
4342adantrr 717 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → 𝑦 / 𝑥𝐵 ≠ ∅)
4443ad2antrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 ≠ ∅)
4544neneqd 2945 . . . . . 6 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → ¬ 𝑦 / 𝑥𝐵 = ∅)
4635, 45condan 818 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) → 𝑦 = 𝑧)
4746ex 412 . . . 4 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧))
4847ralrimivva 3202 . . 3 (𝜑 → ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧))
4915, 48jca 511 . 2 (𝜑 → (∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧)))
50 disjf1.f . . . 4 𝐹 = (𝑥𝐴𝐵)
51 nfcv 2905 . . . . 5 𝑦𝐵
5251, 4, 10cbvmpt 5253 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
5350, 52eqtri 2765 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
54 csbeq1 3902 . . 3 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
5553, 54f1mpt 7281 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧)))
5649, 55sylibr 234 1 (𝜑𝐹:𝐴1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wnf 1783  wcel 2108  wne 2940  wral 3061  csb 3899  cin 3950  c0 4333  Disj wdisj 5110  cmpt 5225  1-1wf1 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569
This theorem is referenced by:  disjf1o  45196  meadjiunlem  46480
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