Theorem invdisjrab 5016

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 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 2899	. . . . . 6 ⊢ Ⅎ𝑥𝑧
2		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝐵
3		nfcsb1v 3814	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶
4	3	nfeq1 2914	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 = 𝑦
5		csbeq1a 3804	. . . . . . 7 ⊢ (𝑥 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑥⦌𝐶)
6	5	eqeq1d 2740	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝐶 = 𝑦 ↔ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦))
7	1, 2, 4, 6	elrabf 3584	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} ↔ (𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑥⦌𝐶 = 𝑦))
8		ax-1 6	. . . . 5 ⊢ (⦋𝑧 / 𝑥⦌𝐶 = 𝑦 → (𝑦 ∈ 𝐴 → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦))
9	7, 8	simplbiim 508	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} → (𝑦 ∈ 𝐴 → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦))
10	9	impcom 411	. . 3 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}) → ⦋𝑧 / 𝑥⦌𝐶 = 𝑦)
11	10	rgen2 3115	. 2 ⊢ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦
12		invdisj 5014	. 2 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}⦋𝑧 / 𝑥⦌𝐶 = 𝑦 → Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦})
13	11, 12	ax-mp 5	1 ⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3053  {crab 3057  ⦋csb 3790  Disj wdisj 4995

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-disj 4996

This theorem is referenced by:  disjwrdpfx  14151