Theorem en3d 3304

Description: Equinumerosity inference from an implicit one-to-one onto function.

Hypotheses

Ref	Expression
en3d.1	⊢ (φ → A ∈ V)
en3d.2	⊢ (φ → (x ∈ A → C ∈ B))
en3d.3	⊢ (φ → (y ∈ B → D ∈ A))
en3d.4	⊢ (φ → ((x ∈ A ∧ y ∈ B) → (x = D ↔ y = C)))

Assertion

Ref	Expression
en3d	⊢ (φ → A ≈ B)

Distinct variable group(s):   x,y,A   x,B,y   y,C   x,D   φ,x,y

Proof of Theorem en3d

Step	Hyp	Ref	Expression
1		en3d.1	. 2 ⊢ (φ → A ∈ V)
2		en3d.2	. . 3 ⊢ (φ → (x ∈ A → C ∈ B))
3		elisset 1354	. . 3 ⊢ (C ∈ B → C ∈ V)
4	2, 3	syl6 23	. 2 ⊢ (φ → (x ∈ A → C ∈ V))
5		en3d.3	. . 3 ⊢ (φ → (y ∈ B → D ∈ A))
6		elisset 1354	. . 3 ⊢ (D ∈ A → D ∈ V)
7	5, 6	syl6 23	. 2 ⊢ (φ → (y ∈ B → D ∈ V))
8		eleq1a 1158	. . . . . . 7 ⊢ (C ∈ B → (y = C → y ∈ B))
9	2, 8	syl6 23	. . . . . 6 ⊢ (φ → (x ∈ A → (y = C → y ∈ B)))
10	9	imp32 281	. . . . 5 ⊢ ((φ ∧ (x ∈ A ∧ y = C)) → y ∈ B)
11		en3d.4	. . . . . . . . . 10 ⊢ (φ → ((x ∈ A ∧ y ∈ B) → (x = D ↔ y = C)))
12	11	imp 277	. . . . . . . . 9 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → (x = D ↔ y = C))
13	12	biimpar 325	. . . . . . . 8 ⊢ (((φ ∧ (x ∈ A ∧ y ∈ B)) ∧ y = C) → x = D)
14	13	exp42 300	. . . . . . 7 ⊢ (φ → (x ∈ A → (y ∈ B → (y = C → x = D))))
15	14	com34 36	. . . . . 6 ⊢ (φ → (x ∈ A → (y = C → (y ∈ B → x = D))))
16	15	imp32 281	. . . . 5 ⊢ ((φ ∧ (x ∈ A ∧ y = C)) → (y ∈ B → x = D))
17	10, 16	jcai 237	. . . 4 ⊢ ((φ ∧ (x ∈ A ∧ y = C)) → (y ∈ B ∧ x = D))
18	17	exp 291	. . 3 ⊢ (φ → ((x ∈ A ∧ y = C) → (y ∈ B ∧ x = D)))
19		eleq1a 1158	. . . . . . 7 ⊢ (D ∈ A → (x = D → x ∈ A))
20	5, 19	syl6 23	. . . . . 6 ⊢ (φ → (y ∈ B → (x = D → x ∈ A)))
21	20	imp32 281	. . . . 5 ⊢ ((φ ∧ (y ∈ B ∧ x = D)) → x ∈ A)
22	12	biimpa 324	. . . . . . . . 9 ⊢ (((φ ∧ (x ∈ A ∧ y ∈ B)) ∧ x = D) → y = C)
23	22	exp42 300	. . . . . . . 8 ⊢ (φ → (x ∈ A → (y ∈ B → (x = D → y = C))))
24	23	com23 32	. . . . . . 7 ⊢ (φ → (y ∈ B → (x ∈ A → (x = D → y = C))))
25	24	com34 36	. . . . . 6 ⊢ (φ → (y ∈ B → (x = D → (x ∈ A → y = C))))
26	25	imp32 281	. . . . 5 ⊢ ((φ ∧ (y ∈ B ∧ x = D)) → (x ∈ A → y = C))
27	21, 26	jcai 237	. . . 4 ⊢ ((φ ∧ (y ∈ B ∧ x = D)) → (x ∈ A ∧ y = C))
28	27	exp 291	. . 3 ⊢ (φ → ((y ∈ B ∧ x = D) → (x ∈ A ∧ y = C)))
29	18, 28	impbid 397	. 2 ⊢ (φ → ((x ∈ A ∧ y = C) ↔ (y ∈ B ∧ x = D)))
30	1, 4, 7, 29	en2d 3303	1 ⊢ (φ → A ≈ B)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054   ≈ cen 3271

This theorem is referenced by:  en3 3306  fundmen 3333  mapunen 3397  ssenen 3399

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274

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