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Theorem disjabrexf 28778
Description: Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.) (Revised by Thierry Arnoux, 9-Mar-2017.)
Hypothesis
Ref Expression
disjabrexf.1 𝑥𝐴
Assertion
Ref Expression
disjabrexf (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem disjabrexf
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfdisj1 4566 . . . 4 𝑥Disj 𝑥𝐴 𝐵
2 nfcv 2751 . . . . 5 𝑥𝑦
3 disjabrexf.1 . . . . . . . . . . 11 𝑥𝐴
43nfcri 2745 . . . . . . . . . 10 𝑥 𝑖𝐴
5 nfcsb1v 3515 . . . . . . . . . . 11 𝑥𝑖 / 𝑥𝐵
65nfcri 2745 . . . . . . . . . 10 𝑥 𝑗𝑖 / 𝑥𝐵
74, 6nfan 1816 . . . . . . . . 9 𝑥(𝑖𝐴𝑗𝑖 / 𝑥𝐵)
87nfab 2755 . . . . . . . 8 𝑥{𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)}
98nfuni 4378 . . . . . . 7 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)}
109nfcsb1 3514 . . . . . 6 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵
1110nfeq1 2764 . . . . 5 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦
122, 11nfral 2929 . . . 4 𝑥𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦
13 eqeq2 2621 . . . . 5 (𝑦 = 𝐵 → ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵))
1413raleqbi1dv 3123 . . . 4 (𝑦 = 𝐵 → (∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦 ↔ ∀𝑗𝐵 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵))
15 vex 3176 . . . . 5 𝑦 ∈ V
1615a1i 11 . . . 4 (Disj 𝑥𝐴 𝐵𝑦 ∈ V)
17 simplll 794 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → Disj 𝑥𝐴 𝐵)
18 simpllr 795 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑥𝐴)
19 simprl 790 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑖𝐴)
20 simplr 788 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑗𝐵)
21 simprr 792 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑗𝑖 / 𝑥𝐵)
22 csbeq1a 3508 . . . . . . . . . . . . . 14 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
233, 5, 22disjif2 28776 . . . . . . . . . . . . 13 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑖𝐴) ∧ (𝑗𝐵𝑗𝑖 / 𝑥𝐵)) → 𝑥 = 𝑖)
2417, 18, 19, 20, 21, 23syl122anc 1327 . . . . . . . . . . . 12 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑥 = 𝑖)
25 simpr 476 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑥 = 𝑖)
26 simpllr 795 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑥𝐴)
2725, 26eqeltrrd 2689 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑖𝐴)
28 simplr 788 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑗𝐵)
2922eleq2d 2673 . . . . . . . . . . . . . . 15 (𝑥 = 𝑖 → (𝑗𝐵𝑗𝑖 / 𝑥𝐵))
3025, 29syl 17 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → (𝑗𝐵𝑗𝑖 / 𝑥𝐵))
3128, 30mpbid 221 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑗𝑖 / 𝑥𝐵)
3227, 31jca 553 . . . . . . . . . . . 12 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → (𝑖𝐴𝑗𝑖 / 𝑥𝐵))
3324, 32impbida 873 . . . . . . . . . . 11 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → ((𝑖𝐴𝑗𝑖 / 𝑥𝐵) ↔ 𝑥 = 𝑖))
34 equcom 1932 . . . . . . . . . . 11 (𝑥 = 𝑖𝑖 = 𝑥)
3533, 34syl6bb 275 . . . . . . . . . 10 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → ((𝑖𝐴𝑗𝑖 / 𝑥𝐵) ↔ 𝑖 = 𝑥))
3635abbidv 2728 . . . . . . . . 9 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑖𝑖 = 𝑥})
37 df-sn 4126 . . . . . . . . 9 {𝑥} = {𝑖𝑖 = 𝑥}
3836, 37syl6eqr 2662 . . . . . . . 8 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑥})
3938unieqd 4382 . . . . . . 7 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑥})
40 vex 3176 . . . . . . . 8 𝑥 ∈ V
4140unisn 4387 . . . . . . 7 {𝑥} = 𝑥
4239, 41syl6eq 2660 . . . . . 6 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥)
43 csbeq1 3502 . . . . . . 7 ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
44 csbid 3507 . . . . . . 7 𝑥 / 𝑥𝐵 = 𝐵
4543, 44syl6eq 2660 . . . . . 6 ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
4642, 45syl 17 . . . . 5 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
4746ralrimiva 2949 . . . 4 ((Disj 𝑥𝐴 𝐵𝑥𝐴) → ∀𝑗𝐵 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
481, 12, 14, 16, 47elabreximd 28732 . . 3 ((Disj 𝑥𝐴 𝐵𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → ∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦)
4948ralrimiva 2949 . 2 (Disj 𝑥𝐴 𝐵 → ∀𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦)
50 invdisj 4571 . 2 (∀𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
5149, 50syl 17 1 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  wnfc 2738  wral 2896  wrex 2897  Vcvv 3173  csb 3499  {csn 4125   cuni 4372  Disj wdisj 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rmo 2904  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-disj 4554
This theorem is referenced by:  measvunilem  29602
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