Theorem invdisjrab 5059

Description: The restricted class abstractions {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} for distinct 𝑦 ∈ 𝐴 are disjoint. (Contributed by AV, 6-May-2020.) (Proof shortened by GG, 26-Jan-2024.)

Assertion

Ref	Expression
invdisjrab	⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 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.

Step	Hyp	Ref	Expression
1		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑥𝑧
2		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑥𝐵
3		nfcsb1v 3855	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶
4	3	nfeq1 2916	. . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 = 𝑦
5		csbeq1a 3845	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑥⦌𝐶)
6	5	eqeq1d 2741	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐶 = 𝑦 ↔ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦))
7	1, 2, 4, 6	elrabf 3626	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} ↔ (𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦))
8		simprr 778	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)
9	7, 8	sylan2b 600	. . 3 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}) → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)
10	9	rgen2 3179	. 2 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦
11		invdisj 5058	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦 → Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦})
12	10, 11	ax-mp 5	1 ⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391  ⦋csb 3831  Disj wdisj 5039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rmo 3344  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-disj 5040

This theorem is referenced by:  disjwrdpfx  14653