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Theorem disjf1 43391
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 1917 . . . . . . 7 𝑥 𝑦𝐴
31, 2nfan 1902 . . . . . 6 𝑥(𝜑𝑦𝐴)
4 nfcsb1v 3880 . . . . . . 7 𝑥𝑦 / 𝑥𝐵
5 nfcv 2907 . . . . . . 7 𝑥𝑉
64, 5nfel 2921 . . . . . 6 𝑥𝑦 / 𝑥𝐵𝑉
73, 6nfim 1899 . . . . 5 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)
8 eleq1w 2820 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98anbi2d 629 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
10 csbeq1a 3869 . . . . . . 7 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1110eleq1d 2822 . . . . . 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 2233 . . . 4 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)
1514ralrimiva 3143 . . 3 (𝜑 → ∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉)
16 inidm 4178 . . . . . . . . 9 (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = 𝑦 / 𝑥𝐵
1716eqcomi 2745 . . . . . . . 8 𝑦 / 𝑥𝐵 = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵)
1817a1i 11 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵))
19 ineq2 4166 . . . . . . . 8 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵))
2019ad2antlr 725 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵))
21 disjf1.dj . . . . . . . . . 10 (𝜑Disj 𝑥𝐴 𝐵)
22 nfcv 2907 . . . . . . . . . . 11 𝑤𝐵
23 nfcsb1v 3880 . . . . . . . . . . 11 𝑥𝑤 / 𝑥𝐵
24 csbeq1a 3869 . . . . . . . . . . 11 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
2522, 23, 24cbvdisj 5080 . . . . . . . . . 10 (Disj 𝑥𝐴 𝐵Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
2621, 25sylib 217 . . . . . . . . 9 (𝜑Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
2726ad3antrrr 728 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
28 simpllr 774 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦𝐴𝑧𝐴))
29 neqne 2951 . . . . . . . . 9 𝑦 = 𝑧𝑦𝑧)
3029adantl 482 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦𝑧)
31 csbeq1 3858 . . . . . . . . 9 (𝑤 = 𝑦𝑤 / 𝑥𝐵 = 𝑦 / 𝑥𝐵)
32 csbeq1 3858 . . . . . . . . 9 (𝑤 = 𝑧𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3331, 32disji2 5087 . . . . . . . 8 ((Disj 𝑤𝐴 𝑤 / 𝑥𝐵 ∧ (𝑦𝐴𝑧𝐴) ∧ 𝑦𝑧) → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅)
3427, 28, 30, 33syl3anc 1371 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅)
3518, 20, 343eqtrd 2780 . . . . . 6 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 = ∅)
36 nfcv 2907 . . . . . . . . . . . 12 𝑥
374, 36nfne 3045 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵 ≠ ∅
383, 37nfim 1899 . . . . . . . . . 10 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)
3910neeq1d 3003 . . . . . . . . . . 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 2233 . . . . . . . . 9 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)
4342adantrr 715 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → 𝑦 / 𝑥𝐵 ≠ ∅)
4443ad2antrr 724 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 ≠ ∅)
4544neneqd 2948 . . . . . 6 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → ¬ 𝑦 / 𝑥𝐵 = ∅)
4635, 45condan 816 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) → 𝑦 = 𝑧)
4746ex 413 . . . 4 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧))
4847ralrimivva 3197 . . 3 (𝜑 → ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧))
4915, 48jca 512 . 2 (𝜑 → (∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧)))
50 disjf1.f . . . 4 𝐹 = (𝑥𝐴𝐵)
51 nfcv 2907 . . . . 5 𝑦𝐵
5251, 4, 10cbvmpt 5216 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
5350, 52eqtri 2764 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
54 csbeq1 3858 . . 3 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
5553, 54f1mpt 7208 . 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 396   = wceq 1541  wnf 1785  wcel 2106  wne 2943  wral 3064  csb 3855  cin 3909  c0 4282  Disj wdisj 5070  cmpt 5188  1-1wf1 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fv 6504
This theorem is referenced by:  disjf1o  43400  meadjiunlem  44696
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