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Theorem disjf1 38843
 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 1840 . . . . . . 7 𝑥 𝑦𝐴
31, 2nfan 1825 . . . . . 6 𝑥(𝜑𝑦𝐴)
4 nfcsb1v 3530 . . . . . . 7 𝑥𝑦 / 𝑥𝐵
5 nfcv 2761 . . . . . . 7 𝑥𝑉
64, 5nfel 2773 . . . . . 6 𝑥𝑦 / 𝑥𝐵𝑉
73, 6nfim 1822 . . . . 5 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)
8 eleq1 2686 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
98anbi2d 739 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
10 csbeq1a 3523 . . . . . . 7 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1110eleq1d 2683 . . . . . 6 (𝑥 = 𝑦 → (𝐵𝑉𝑦 / 𝑥𝐵𝑉))
129, 11imbi12d 334 . . . . 5 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝐵𝑉) ↔ ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)))
13 disjf1.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
147, 12, 13chvar 2261 . . . 4 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵𝑉)
1514ralrimiva 2960 . . 3 (𝜑 → ∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉)
16 inidm 3800 . . . . . . . . 9 (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = 𝑦 / 𝑥𝐵
1716eqcomi 2630 . . . . . . . 8 𝑦 / 𝑥𝐵 = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵)
1817a1i 11 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 = (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵))
19 ineq2 3786 . . . . . . . 8 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵 → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵))
2019ad2antlr 762 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 / 𝑥𝐵𝑦 / 𝑥𝐵) = (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵))
21 disjf1.dj . . . . . . . . . 10 (𝜑Disj 𝑥𝐴 𝐵)
22 nfcv 2761 . . . . . . . . . . 11 𝑤𝐵
23 nfcsb1v 3530 . . . . . . . . . . 11 𝑥𝑤 / 𝑥𝐵
24 csbeq1a 3523 . . . . . . . . . . 11 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
2522, 23, 24cbvdisj 4593 . . . . . . . . . 10 (Disj 𝑥𝐴 𝐵Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
2621, 25sylib 208 . . . . . . . . 9 (𝜑Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
2726ad3antrrr 765 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → Disj 𝑤𝐴 𝑤 / 𝑥𝐵)
28 simpllr 798 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦𝐴𝑧𝐴))
29 neqne 2798 . . . . . . . . 9 𝑦 = 𝑧𝑦𝑧)
3029adantl 482 . . . . . . . 8 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦𝑧)
31 csbeq1 3517 . . . . . . . . 9 (𝑤 = 𝑦𝑤 / 𝑥𝐵 = 𝑦 / 𝑥𝐵)
32 csbeq1 3517 . . . . . . . . 9 (𝑤 = 𝑧𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
3331, 32disji2 4599 . . . . . . . 8 ((Disj 𝑤𝐴 𝑤 / 𝑥𝐵 ∧ (𝑦𝐴𝑧𝐴) ∧ 𝑦𝑧) → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅)
3427, 28, 30, 33syl3anc 1323 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅)
3518, 20, 343eqtrd 2659 . . . . . 6 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 = ∅)
36 nfcv 2761 . . . . . . . . . . . 12 𝑥
374, 36nfne 2890 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵 ≠ ∅
383, 37nfim 1822 . . . . . . . . . 10 𝑥((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)
3910neeq1d 2849 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝐵 ≠ ∅ ↔ 𝑦 / 𝑥𝐵 ≠ ∅))
409, 39imbi12d 334 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → 𝐵 ≠ ∅) ↔ ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)))
41 disjf1.n0 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ≠ ∅)
4238, 40, 41chvar 2261 . . . . . . . . 9 ((𝜑𝑦𝐴) → 𝑦 / 𝑥𝐵 ≠ ∅)
4342adantrr 752 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → 𝑦 / 𝑥𝐵 ≠ ∅)
4443ad2antrr 761 . . . . . . 7 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → 𝑦 / 𝑥𝐵 ≠ ∅)
4544neneqd 2795 . . . . . 6 ((((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) ∧ ¬ 𝑦 = 𝑧) → ¬ 𝑦 / 𝑥𝐵 = ∅)
4635, 45condan 834 . . . . 5 (((𝜑 ∧ (𝑦𝐴𝑧𝐴)) ∧ 𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵) → 𝑦 = 𝑧)
4746ex 450 . . . 4 ((𝜑 ∧ (𝑦𝐴𝑧𝐴)) → (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧))
4847ralrimivva 2965 . . 3 (𝜑 → ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧))
4915, 48jca 554 . 2 (𝜑 → (∀𝑦𝐴 𝑦 / 𝑥𝐵𝑉 ∧ ∀𝑦𝐴𝑧𝐴 (𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵𝑦 = 𝑧)))
50 disjf1.f . . . 4 𝐹 = (𝑥𝐴𝐵)
51 nfcv 2761 . . . . 5 𝑦𝐵
5251, 4, 10cbvmpt 4709 . . . 4 (𝑥𝐴𝐵) = (𝑦𝐴𝑦 / 𝑥𝐵)
5350, 52eqtri 2643 . . 3 𝐹 = (𝑦𝐴𝑦 / 𝑥𝐵)
54 csbeq1 3517 . . 3 (𝑦 = 𝑧𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
5553, 54f1mpt 6472 . 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 384   = wceq 1480  Ⅎwnf 1705   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ⦋csb 3514   ∩ cin 3554  ∅c0 3891  Disj wdisj 4583   ↦ cmpt 4673  –1-1→wf1 5844 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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867 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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fv 5855 This theorem is referenced by:  disjf1o  38852  meadjiunlem  39989
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