Theorem invdisj 4023

Description: If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all 𝑦 ∈ 𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥 ∈ 𝐴 are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)

Assertion

Ref	Expression
invdisj	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → Disj 𝑥 ∈ 𝐴 𝐵)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem invdisj

Step	Hyp	Ref	Expression
1		nfra2xy 2536	. . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥
2		df-ral 2477	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥))
3		rsp 2541	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → (𝑦 ∈ 𝐵 → 𝐶 = 𝑥))
4		eqcom 2195	. . . . . . . . 9 ⊢ (𝐶 = 𝑥 ↔ 𝑥 = 𝐶)
5	3, 4	imbitrdi 161	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐶))
6	5	imim2i 12	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐶)))
7	6	impd 254	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶))
8	7	alimi 1466	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶))
9	2, 8	sylbi 121	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶))
10		mo2icl 2939	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
11	9, 10	syl 14	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
12	1, 11	alrimi 1533	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
13		dfdisj2 4008	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
14	12, 13	sylibr 134	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → Disj 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362   = wceq 1364  ∃*wmo 2043   ∈ wcel 2164  ∀wral 2472  Disj wdisj 4006

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rmo 2480  df-v 2762  df-disj 4007

This theorem is referenced by:  phisum  12378  lgsquadlem1  15191