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Theorem disjabrexf 29264
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 4601 . . . 4 𝑥Disj 𝑥𝐴 𝐵
2 nfcv 2761 . . . . 5 𝑥𝑦
3 disjabrexf.1 . . . . . . . . . . 11 𝑥𝐴
43nfcri 2755 . . . . . . . . . 10 𝑥 𝑖𝐴
5 nfcsb1v 3534 . . . . . . . . . . 11 𝑥𝑖 / 𝑥𝐵
65nfcri 2755 . . . . . . . . . 10 𝑥 𝑗𝑖 / 𝑥𝐵
74, 6nfan 1825 . . . . . . . . 9 𝑥(𝑖𝐴𝑗𝑖 / 𝑥𝐵)
87nfab 2765 . . . . . . . 8 𝑥{𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)}
98nfuni 4413 . . . . . . 7 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)}
109nfcsb1 3533 . . . . . 6 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵
1110nfeq1 2774 . . . . 5 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦
122, 11nfral 2940 . . . 4 𝑥𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦
13 eqeq2 2632 . . . . 5 (𝑦 = 𝐵 → ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵))
1413raleqbi1dv 3138 . . . 4 (𝑦 = 𝐵 → (∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦 ↔ ∀𝑗𝐵 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵))
15 vex 3192 . . . . 5 𝑦 ∈ V
1615a1i 11 . . . 4 (Disj 𝑥𝐴 𝐵𝑦 ∈ V)
17 simplll 797 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → Disj 𝑥𝐴 𝐵)
18 simpllr 798 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑥𝐴)
19 simprl 793 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑖𝐴)
20 simplr 791 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑗𝐵)
21 simprr 795 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑗𝑖 / 𝑥𝐵)
22 csbeq1a 3527 . . . . . . . . . . . . . 14 (𝑥 = 𝑖𝐵 = 𝑖 / 𝑥𝐵)
233, 5, 22disjif2 29262 . . . . . . . . . . . . 13 ((Disj 𝑥𝐴 𝐵 ∧ (𝑥𝐴𝑖𝐴) ∧ (𝑗𝐵𝑗𝑖 / 𝑥𝐵)) → 𝑥 = 𝑖)
2417, 18, 19, 20, 21, 23syl122anc 1332 . . . . . . . . . . . 12 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)) → 𝑥 = 𝑖)
25 simpr 477 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑥 = 𝑖)
26 simpllr 798 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑥𝐴)
2725, 26eqeltrrd 2699 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑖𝐴)
28 simplr 791 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑗𝐵)
2922eleq2d 2684 . . . . . . . . . . . . . . 15 (𝑥 = 𝑖 → (𝑗𝐵𝑗𝑖 / 𝑥𝐵))
3025, 29syl 17 . . . . . . . . . . . . . 14 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → (𝑗𝐵𝑗𝑖 / 𝑥𝐵))
3128, 30mpbid 222 . . . . . . . . . . . . 13 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → 𝑗𝑖 / 𝑥𝐵)
3227, 31jca 554 . . . . . . . . . . . 12 ((((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) ∧ 𝑥 = 𝑖) → (𝑖𝐴𝑗𝑖 / 𝑥𝐵))
3324, 32impbida 876 . . . . . . . . . . 11 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → ((𝑖𝐴𝑗𝑖 / 𝑥𝐵) ↔ 𝑥 = 𝑖))
34 equcom 1942 . . . . . . . . . . 11 (𝑥 = 𝑖𝑖 = 𝑥)
3533, 34syl6bb 276 . . . . . . . . . 10 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → ((𝑖𝐴𝑗𝑖 / 𝑥𝐵) ↔ 𝑖 = 𝑥))
3635abbidv 2738 . . . . . . . . 9 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑖𝑖 = 𝑥})
37 df-sn 4154 . . . . . . . . 9 {𝑥} = {𝑖𝑖 = 𝑥}
3836, 37syl6eqr 2673 . . . . . . . 8 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑥})
3938unieqd 4417 . . . . . . 7 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = {𝑥})
40 vex 3192 . . . . . . . 8 𝑥 ∈ V
4140unisn 4422 . . . . . . 7 {𝑥} = 𝑥
4239, 41syl6eq 2671 . . . . . 6 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥)
43 csbeq1 3521 . . . . . . 7 ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑥 / 𝑥𝐵)
44 csbid 3526 . . . . . . 7 𝑥 / 𝑥𝐵 = 𝐵
4543, 44syl6eq 2671 . . . . . 6 ( {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} = 𝑥 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
4642, 45syl 17 . . . . 5 (((Disj 𝑥𝐴 𝐵𝑥𝐴) ∧ 𝑗𝐵) → {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
4746ralrimiva 2961 . . . 4 ((Disj 𝑥𝐴 𝐵𝑥𝐴) → ∀𝑗𝐵 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝐵)
481, 12, 14, 16, 47elabreximd 29218 . . 3 ((Disj 𝑥𝐴 𝐵𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → ∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦)
4948ralrimiva 2961 . 2 (Disj 𝑥𝐴 𝐵 → ∀𝑦 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}∀𝑗𝑦 {𝑖 ∣ (𝑖𝐴𝑗𝑖 / 𝑥𝐵)} / 𝑥𝐵 = 𝑦)
50 invdisj 4606 . 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 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wnfc 2748  wral 2907  wrex 2908  Vcvv 3189  csb 3518  {csn 4153   cuni 4407  Disj wdisj 4588
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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-rmo 2915  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-nul 3897  df-sn 4154  df-pr 4156  df-uni 4408  df-disj 4589
This theorem is referenced by:  measvunilem  30080
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