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Theorem invdisjrab 4791
 Description: The restricted class abstractions {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} for distinct 𝑦 ∈ 𝐴 are disjoint. (Contributed by AV, 6-May-2020.)
Assertion
Ref Expression
invdisjrab Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
Distinct variable groups:   𝑥,𝐵   𝑦,𝐶   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝐶(𝑥)

Proof of Theorem invdisjrab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2902 . . . . . 6 𝑥𝑧
2 nfcv 2902 . . . . . 6 𝑥𝐵
3 nfcsb1v 3690 . . . . . . 7 𝑥𝑧 / 𝑥𝐶
43nfeq1 2916 . . . . . 6 𝑥𝑧 / 𝑥𝐶 = 𝑦
5 csbeq1a 3683 . . . . . . 7 (𝑥 = 𝑧𝐶 = 𝑧 / 𝑥𝐶)
65eqeq1d 2762 . . . . . 6 (𝑥 = 𝑧 → (𝐶 = 𝑦𝑧 / 𝑥𝐶 = 𝑦))
71, 2, 4, 6elrabf 3500 . . . . 5 (𝑧 ∈ {𝑥𝐵𝐶 = 𝑦} ↔ (𝑧𝐵𝑧 / 𝑥𝐶 = 𝑦))
8 ax-1 6 . . . . 5 (𝑧 / 𝑥𝐶 = 𝑦 → (𝑦𝐴𝑧 / 𝑥𝐶 = 𝑦))
97, 8simplbiim 661 . . . 4 (𝑧 ∈ {𝑥𝐵𝐶 = 𝑦} → (𝑦𝐴𝑧 / 𝑥𝐶 = 𝑦))
109impcom 445 . . 3 ((𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}) → 𝑧 / 𝑥𝐶 = 𝑦)
1110rgen2 3113 . 2 𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}𝑧 / 𝑥𝐶 = 𝑦
12 invdisj 4790 . 2 (∀𝑦𝐴𝑧 ∈ {𝑥𝐵𝐶 = 𝑦}𝑧 / 𝑥𝐶 = 𝑦Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦})
1311, 12ax-mp 5 1 Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054  ⦋csb 3674  Disj wdisj 4772 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-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-ral 3055  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-disj 4773 This theorem is referenced by:  disjxwrd  13655  disjwrdpfx  41918
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