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Theorem disjabrex 30318
 Description: Rewriting a disjoint collection into a partition of its image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Assertion
Ref Expression
disjabrex (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjabrex
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfdisj1 5018 . . . 4 𝑥Disj 𝑥𝐴 𝐵
2 nfcv 2974 . . . . 5 𝑥𝑦
3 nfv 1916 . . . . . . . . . 10 𝑥 𝑖𝐴
4 nfcsb1v 3881 . . . . . . . . . . 11 𝑥𝑖 / 𝑥𝐵
54nfcri 2968 . . . . . . . . . 10 𝑥 𝑗𝑖 / 𝑥𝐵
63, 5nfan 1901 . . . . . . . . 9 𝑥(𝑖𝐴𝑗𝑖 / 𝑥𝐵)
76nfab 2980 . . . . . . . 8 𝑥{𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)}
87nfuni 4818 . . . . . . 7 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)}
98nfcsb1 3880 . . . . . 6 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵
109nfeq1 2989 . . . . 5 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦
112, 10nfralw 3213 . . . 4 𝑥𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦
12 eqeq2 2833 . . . . 5 (𝑦 = 𝐵 → ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵))
1312raleqbi1dv 3388 . . . 4 (𝑦 = 𝐵 → (∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦 ↔ ∀𝑗𝐵 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵))
14 vex 3474 . . . . 5 𝑦 ∈ V
1514a1i 11 . . . 4 (Disj 𝑥𝐴 𝐵𝑦 ∈ V)
16 simplll 774 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → Disj 𝑥𝐴 𝐵)
17 simpllr 775 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑥𝐴)
18 simprl 770 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑖𝐴)
19 simplr 768 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑗𝐵)
20 simprr 772 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑗𝑖 / 𝑥𝐵)
21 csbeq1a 3871 . . . . . . . . . . . . . 14 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
224, 21disjif 30314 . . . . . . . . . . . . 13 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑖𝐴) ∧ (𝑗𝐵𝑗𝑖 / 𝑥𝐵)) → 𝑥 = 𝑖)
2316, 17, 18, 19, 20, 22syl122anc 1376 . . . . . . . . . . . 12 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑥 = 𝑖)
24 simpr 488 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑥 = 𝑖)
25 simpllr 775 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑥𝐴)
2624, 25eqeltrrd 2913 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑖𝐴)
27 simplr 768 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑗𝐵)
2821eleq2d 2897 . . . . . . . . . . . . . . 15 (𝑥 = 𝑖 → (𝑗𝐵𝑗𝑖 / 𝑥𝐵))
2924, 28syl 17 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → (𝑗𝐵𝑗𝑖 / 𝑥𝐵))
3027, 29mpbid 235 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑗𝑖 / 𝑥𝐵)
3126, 30jca 515 . . . . . . . . . . . 12 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → (𝑖𝐴𝑗𝑖 / 𝑥𝐵))
3223, 31impbida 800 . . . . . . . . . . 11 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → ((𝑖𝐴𝑗𝑖 / 𝑥𝐵) ↔ 𝑥 = 𝑖))
33 equcom 2026 . . . . . . . . . . 11 (𝑥 = 𝑖𝑖 = 𝑥)
3432, 33syl6bb 290 . . . . . . . . . 10 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → ((𝑖𝐴𝑗𝑖 / 𝑥𝐵) ↔ 𝑖 = 𝑥))
3534abbidv 2885 . . . . . . . . 9 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑖𝑖 = 𝑥})
36 df-sn 4541 . . . . . . . . 9 {𝑥} = {𝑖𝑖 = 𝑥}
3735, 36syl6eqr 2874 . . . . . . . 8 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑥})
3837unieqd 4825 . . . . . . 7 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑥})
39 vex 3474 . . . . . . . 8 𝑥 ∈ V
4039unisn 4831 . . . . . . 7 {𝑥} = 𝑥
4138, 40syl6eq 2872 . . . . . 6 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥)
42 csbeq1 3860 . . . . . . 7 ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
43 csbid 3870 . . . . . . 7 𝑥 / 𝑥𝐵 = 𝐵
4442, 43syl6eq 2872 . . . . . 6 ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
4541, 44syl 17 . . . . 5 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
4645ralrimiva 3170 . . . 4 ((Disj 𝑥𝐴 𝐵𝑥𝐴) → ∀𝑗𝐵 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
471, 11, 13, 15, 46elabreximd 30255 . . 3 ((Disj 𝑥𝐴 𝐵𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → ∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦)
4847ralrimiva 3170 . 2 (Disj 𝑥𝐴 𝐵 → ∀𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦)
49 invdisj 5023 . 2 (∀𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
5048, 49syl 17 1 (Disj 𝑥𝐴 𝐵Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  {cab 2799  ∀wral 3126  ∃wrex 3127  Vcvv 3471  ⦋csb 3857  {csn 4540  ∪ cuni 4811  Disj wdisj 5004 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-sn 4541  df-pr 4543  df-uni 4812  df-disj 5005 This theorem is referenced by:  disjrnmpt  30321
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