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Theorem disjf1 39868
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 1992 . . . . . . 7 𝑥 𝑦𝐴
31, 2nfan 1977 . . . . . 6 𝑥(𝜑𝑦𝐴)
4 nfcsb1v 3690 . . . . . . 7 𝑥𝑦 / 𝑥𝐵
5 nfcv 2902 . . . . . . 7 𝑥𝑉
64, 5nfel 2915 . . . . . 6 𝑥𝑦 / 𝑥𝐵𝑉
73, 6nfim 1974 . . . . 5 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)
8 eleq1w 2822 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98anbi2d 742 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
10 csbeq1a 3683 . . . . . . 7 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1110eleq1d 2824 . . . . . 6 (𝑥 = 𝑦 → (𝐵𝑉𝑦 / 𝑥𝐵𝑉))
129, 11imbi12d 333 . . . . 5 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝐵𝑉) ↔ ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)))
13 disjf1.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
147, 12, 13chvar 2407 . . . 4 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)
1514ralrimiva 3104 . . 3 (𝜑 → ∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉)
16 inidm 3965 . . . . . . . . 9 (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = 𝑦 / 𝑥𝐵
1716eqcomi 2769 . . . . . . . 8 𝑦 / 𝑥𝐵 = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵)
1817a1i 11 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵))
19 ineq2 3951 . . . . . . . 8 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵))
2019ad2antlr 765 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵))
21 disjf1.dj . . . . . . . . . 10 (𝜑Disj 𝑥𝐴 𝐵)
22 nfcv 2902 . . . . . . . . . . 11 𝑤𝐵
23 nfcsb1v 3690 . . . . . . . . . . 11 𝑥𝑤 / 𝑥𝐵
24 csbeq1a 3683 . . . . . . . . . . 11 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
2522, 23, 24cbvdisj 4782 . . . . . . . . . 10 (Disj 𝑥𝐴 𝐵Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
2621, 25sylib 208 . . . . . . . . 9 (𝜑Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
2726ad3antrrr 768 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
28 simpllr 817 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦𝐴𝑧𝐴))
29 neqne 2940 . . . . . . . . 9 𝑦 = 𝑧𝑦𝑧)
3029adantl 473 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦𝑧)
31 csbeq1 3677 . . . . . . . . 9 (𝑤 = 𝑦𝑤 / 𝑥𝐵 = 𝑦 / 𝑥𝐵)
32 csbeq1 3677 . . . . . . . . 9 (𝑤 = 𝑧𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3331, 32disji2 4788 . . . . . . . 8 ((Disj 𝑤𝐴 𝑤 / 𝑥𝐵 ∧ (𝑦𝐴𝑧𝐴) ∧ 𝑦𝑧) → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅)
3427, 28, 30, 33syl3anc 1477 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅)
3518, 20, 343eqtrd 2798 . . . . . 6 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 = ∅)
36 nfcv 2902 . . . . . . . . . . . 12 𝑥
374, 36nfne 3032 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵 ≠ ∅
383, 37nfim 1974 . . . . . . . . . 10 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)
3910neeq1d 2991 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐵 ≠ ∅ ↔ 𝑦 / 𝑥𝐵 ≠ ∅))
409, 39imbi12d 333 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝐵 ≠ ∅) ↔ ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)))
41 disjf1.n0 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
4238, 40, 41chvar 2407 . . . . . . . . 9 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)
4342adantrr 755 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → 𝑦 / 𝑥𝐵 ≠ ∅)
4443ad2antrr 764 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 ≠ ∅)
4544neneqd 2937 . . . . . 6 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → ¬ 𝑦 / 𝑥𝐵 = ∅)
4635, 45condan 870 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) → 𝑦 = 𝑧)
4746ex 449 . . . 4 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧))
4847ralrimivva 3109 . . 3 (𝜑 → ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧))
4915, 48jca 555 . 2 (𝜑 → (∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧)))
50 disjf1.f . . . 4 𝐹 = (𝑥𝐴𝐵)
51 nfcv 2902 . . . . 5 𝑦𝐵
5251, 4, 10cbvmpt 4901 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
5350, 52eqtri 2782 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
54 csbeq1 3677 . . 3 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
5553, 54f1mpt 6681 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧)))
5649, 55sylibr 224 1 (𝜑𝐹:𝐴1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wnf 1857  wcel 2139  wne 2932  wral 3050  csb 3674  cin 3714  c0 4058  Disj wdisj 4772  cmpt 4881  1-1wf1 6046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fv 6057
This theorem is referenced by:  disjf1o  39877  meadjiunlem  41185
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