Theorem en2d 3303

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

Hypotheses

Ref	Expression
en2d.1	⊢ (φ → A ∈ V)
en2d.2	⊢ (φ → (x ∈ A → C ∈ V))
en2d.3	⊢ (φ → (y ∈ B → D ∈ V))
en2d.4	⊢ (φ → ((x ∈ A ∧ y = C) ↔ (y ∈ B ∧ x = D)))

Assertion

Ref	Expression
en2d	⊢ (φ → A ≈ B)

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

Proof of Theorem en2d

Step	Hyp	Ref	Expression
1		f1oeng 3298	. 2 ⊢ (A ∈ V → ({⟨x, y⟩∣(y ∈ B ∧ x = D)}:A–1-1-onto→B → A ≈ B))
2		en2d.1	. 2 ⊢ (φ → A ∈ V)
3		en2d.2	. . . . . . . 8 ⊢ (φ → (x ∈ A → C ∈ V))
4		eueq 1427	. . . . . . . 8 ⊢ (C ∈ V ↔ ∃!y y = C)
5	3, 4	syl6ib 185	. . . . . . 7 ⊢ (φ → (x ∈ A → ∃!y y = C))
6	5	r19.21aiv 1259	. . . . . 6 ⊢ (φ → ∀x ∈ A ∃!y y = C)
7		cleqid 1102	. . . . . . 7 ⊢ {⟨x, y⟩∣(x ∈ A ∧ y = C)} = {⟨x, y⟩∣(x ∈ A ∧ y = C)}
8	7	fnopabg 2745	. . . . . 6 ⊢ (∀x ∈ A ∃!y y = C ↔ {⟨x, y⟩∣(x ∈ A ∧ y = C)} Fn A)
9	6, 8	sylib 173	. . . . 5 ⊢ (φ → {⟨x, y⟩∣(x ∈ A ∧ y = C)} Fn A)
10		en2d.4	. . . . . . 7 ⊢ (φ → ((x ∈ A ∧ y = C) ↔ (y ∈ B ∧ x = D)))
11	10	biopabdv 2102	. . . . . 6 ⊢ (φ → {⟨x, y⟩∣(x ∈ A ∧ y = C)} = {⟨x, y⟩∣(y ∈ B ∧ x = D)})
12		fneq1 2718	. . . . . 6 ⊢ ({⟨x, y⟩∣(x ∈ A ∧ y = C)} = {⟨x, y⟩∣(y ∈ B ∧ x = D)} → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} Fn A ↔ {⟨x, y⟩∣(y ∈ B ∧ x = D)} Fn A))
13	11, 12	syl 12	. . . . 5 ⊢ (φ → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} Fn A ↔ {⟨x, y⟩∣(y ∈ B ∧ x = D)} Fn A))
14	9, 13	mpbid 170	. . . 4 ⊢ (φ → {⟨x, y⟩∣(y ∈ B ∧ x = D)} Fn A)
15		en2d.3	. . . . . . 7 ⊢ (φ → (y ∈ B → D ∈ V))
16		eueq 1427	. . . . . . 7 ⊢ (D ∈ V ↔ ∃!x x = D)
17	15, 16	syl6ib 185	. . . . . 6 ⊢ (φ → (y ∈ B → ∃!x x = D))
18	17	r19.21aiv 1259	. . . . 5 ⊢ (φ → ∀y ∈ B ∃!x x = D)
19		cnvopab 2632	. . . . . 6 ⊢ ◡{⟨x, y⟩∣(y ∈ B ∧ x = D)} = {⟨y, x⟩∣(y ∈ B ∧ x = D)}
20	19	fnopabg 2745	. . . . 5 ⊢ (∀y ∈ B ∃!x x = D ↔ ◡{⟨x, y⟩∣(y ∈ B ∧ x = D)} Fn B)
21	18, 20	sylib 173	. . . 4 ⊢ (φ → ◡{⟨x, y⟩∣(y ∈ B ∧ x = D)} Fn B)
22	14, 21	jca 236	. . 3 ⊢ (φ → ({⟨x, y⟩∣(y ∈ B ∧ x = D)} Fn A ∧ ◡{⟨x, y⟩∣(y ∈ B ∧ x = D)} Fn B))
23		f1o4 2807	. . 3 ⊢ ({⟨x, y⟩∣(y ∈ B ∧ x = D)}:A–1-1-onto→B ↔ ({⟨x, y⟩∣(y ∈ B ∧ x = D)} Fn A ∧ ◡{⟨x, y⟩∣(y ∈ B ∧ x = D)} Fn B))
24	22, 23	sylibr 175	. 2 ⊢ (φ → {⟨x, y⟩∣(y ∈ B ∧ x = D)}:A–1-1-onto→B)
25	1, 2, 24	sylc 62	1 ⊢ (φ → A ≈ B)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   class class class wbr 2054  {copab 2055  ^◡ccnv 2409   Fn wfn 2417  –1-1-onto→wf1o 2421   ≈ cen 3271

This theorem is referenced by:  en3d 3304  en2 3305

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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