Theorem invdisj 5058

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		nfra2w 3275	. . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥
2		df-ral 3054	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥))
3		rsp 3227	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → (𝑦 ∈ 𝐵 → 𝐶 = 𝑥))
4		eqcom 2746	. . . . . . . . 9 ⊢ (𝐶 = 𝑥 ↔ 𝑥 = 𝐶)
5	3, 4	imbitrdi 252	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐶))
6	5	imim2i 16	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐶)))
7	6	impd 411	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶))
8	7	alimi 1818	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶))
9	2, 8	sylbi 218	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶))
10		mo2icl 3655	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
11	9, 10	syl 17	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
12	1, 11	alrimi 2225	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
13		dfdisj2 5041	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
14	12, 13	sylibr 235	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → Disj 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  ∃*wmo 2541  ∀wral 3053  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-ral 3054  df-rmo 3344  df-v 3433  df-disj 5040

This theorem is referenced by:  invdisjrab  5059  ackbijnn  15784  incexc2  15794  phisum  16752  itg1addlem1  25677  musum  27172  lgsquadlem1  27361  lgsquadlem2  27362  disjabrex  32671  disjabrexf  32672  actfunsnrndisj  34789  poimirlem27  38014