Theorem invdisjrab 4108

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 2386	. . . . 5 ⊢ Ⅎ𝑥𝑧
2		nfcv 2386	. . . . 5 ⊢ Ⅎ𝑥𝐵
3		nfcsb1v 3174	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶
4	3	nfeq1 2396	. . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 = 𝑦
5		csbeq1a 3150	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑥⦌𝐶)
6	5	eqeq1d 2243	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐶 = 𝑦 ↔ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦))
7	1, 2, 4, 6	elrabf 2974	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} ↔ (𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦))
8		simprr 533	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)) → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)
9	7, 8	sylan2b 287	. . 3 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}) → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)
10	9	rgen2 2630	. 2 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦
11		invdisj 4107	. 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 104   = wceq 1398   ∈ wcel 2205  ∀wral 2522  {crab 2526  ⦋csb 3141  Disj wdisj 4090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-disj 4091

This theorem is referenced by:  disjwrdpfx  11417