Theorem fopab2 2891

Description: Functionality of an ordered pair abstraction.

Hypothesis

Ref	Expression
fopab2.1	⊢ F = {⟨x, y⟩∣(x ∈ A ∧ y = C)}

Assertion

Ref	Expression
fopab2	⊢ (∀x ∈ A C ∈ B ↔ F:A–→B)

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

Proof of Theorem fopab2

Step	Hyp	Ref	Expression
1		elisset 1354	. . . . . . 7 ⊢ (C ∈ B → C ∈ V)
2		eueq 1427	. . . . . . 7 ⊢ (C ∈ V ↔ ∃!y y = C)
3	1, 2	sylib 173	. . . . . 6 ⊢ (C ∈ B → ∃!y y = C)
4	3	r19.20si 1254	. . . . 5 ⊢ (∀x ∈ A C ∈ B → ∀x ∈ A ∃!y y = C)
5		fopab2.1	. . . . . 6 ⊢ F = {⟨x, y⟩∣(x ∈ A ∧ y = C)}
6	5	fnopabg 2745	. . . . 5 ⊢ (∀x ∈ A ∃!y y = C ↔ F Fn A)
7	4, 6	sylib 173	. . . 4 ⊢ (∀x ∈ A C ∈ B → F Fn A)
8		hbra1 1237	. . . . . . . . 9 ⊢ (∀x ∈ A C ∈ B → ∀x∀x ∈ A C ∈ B)
9		ax-17 925	. . . . . . . . 9 ⊢ (y ∈ B → ∀x y ∈ B)
10		ra4 1243	. . . . . . . . . 10 ⊢ (∀x ∈ A C ∈ B → (x ∈ A → C ∈ B))
11		eleq1a 1158	. . . . . . . . . . . 12 ⊢ (C ∈ B → (y = C → y ∈ B))
12	11	syl3 18	. . . . . . . . . . 11 ⊢ ((x ∈ A → C ∈ B) → (x ∈ A → (y = C → y ∈ B)))
13	12	imp3a 279	. . . . . . . . . 10 ⊢ ((x ∈ A → C ∈ B) → ((x ∈ A ∧ y = C) → y ∈ B))
14	10, 13	syl 12	. . . . . . . . 9 ⊢ (∀x ∈ A C ∈ B → ((x ∈ A ∧ y = C) → y ∈ B))
15	8, 9, 14	19.23ad 748	. . . . . . . 8 ⊢ (∀x ∈ A C ∈ B → (∃x(x ∈ A ∧ y = C) → y ∈ B))
16		rnopab 2566	. . . . . . . . 9 ⊢ ran {⟨x, y⟩∣(x ∈ A ∧ y = C)} = {y∣∃x(x ∈ A ∧ y = C)}
17	16	cleqabi 1176	. . . . . . . 8 ⊢ (y ∈ ran {⟨x, y⟩∣(x ∈ A ∧ y = C)} ↔ ∃x(x ∈ A ∧ y = C))
18	15, 17	syl5ib 181	. . . . . . 7 ⊢ (∀x ∈ A C ∈ B → (y ∈ ran {⟨x, y⟩∣(x ∈ A ∧ y = C)} → y ∈ B))
19	18	19.21aiv 943	. . . . . 6 ⊢ (∀x ∈ A C ∈ B → ∀y(y ∈ ran {⟨x, y⟩∣(x ∈ A ∧ y = C)} → y ∈ B))
20		hbopab2 2113	. . . . . . . 8 ⊢ (z ∈ {⟨x, y⟩∣(x ∈ A ∧ y = C)} → ∀y z ∈ {⟨x, y⟩∣(x ∈ A ∧ y = C)})
21	20	hbrn 2564	. . . . . . 7 ⊢ (z ∈ ran {⟨x, y⟩∣(x ∈ A ∧ y = C)} → ∀y z ∈ ran {⟨x, y⟩∣(x ∈ A ∧ y = C)})
22		ax-17 925	. . . . . . 7 ⊢ (z ∈ B → ∀y z ∈ B)
23	21, 22	dfss2f 1499	. . . . . 6 ⊢ (ran {⟨x, y⟩∣(x ∈ A ∧ y = C)} ⊆ B ↔ ∀y(y ∈ ran {⟨x, y⟩∣(x ∈ A ∧ y = C)} → y ∈ B))
24	19, 23	sylibr 175	. . . . 5 ⊢ (∀x ∈ A C ∈ B → ran {⟨x, y⟩∣(x ∈ A ∧ y = C)} ⊆ B)
25	5	rneqi 2556	. . . . 5 ⊢ ran F = ran {⟨x, y⟩∣(x ∈ A ∧ y = C)}
26	24, 25	syl5ss 1544	. . . 4 ⊢ (∀x ∈ A C ∈ B → ran F ⊆ B)
27	7, 26	jca 236	. . 3 ⊢ (∀x ∈ A C ∈ B → (F Fn A ∧ ran F ⊆ B))
28		df-f 2434	. . 3 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
29	27, 28	sylibr 175	. 2 ⊢ (∀x ∈ A C ∈ B → F:A–→B)
30		fdm 2756	. . . 4 ⊢ (F:A–→B → dom F = A)
31		dmopab2 2541	. . . . 5 ⊢ (∀x ∈ A ∃y y = C ↔ dom {⟨x, y⟩∣(x ∈ A ∧ y = C)} = A)
32		isset 1351	. . . . . 6 ⊢ (C ∈ V ↔ ∃y y = C)
33	32	biral 1223	. . . . 5 ⊢ (∀x ∈ A C ∈ V ↔ ∀x ∈ A ∃y y = C)
34	5	dmeqi 2532	. . . . . 6 ⊢ dom F = dom {⟨x, y⟩∣(x ∈ A ∧ y = C)}
35	34	cleq1i 1108	. . . . 5 ⊢ (dom F = A ↔ dom {⟨x, y⟩∣(x ∈ A ∧ y = C)} = A)
36	31, 33, 35	3bitr4r 159	. . . 4 ⊢ (dom F = A ↔ ∀x ∈ A C ∈ V)
37	30, 36	sylib 173	. . 3 ⊢ (F:A–→B → ∀x ∈ A C ∈ V)
38		hbopab1 2112	. . . . . 6 ⊢ (z ∈ {⟨x, y⟩∣(x ∈ A ∧ y = C)} → ∀x z ∈ {⟨x, y⟩∣(x ∈ A ∧ y = C)})
39		ax-17 925	. . . . . 6 ⊢ (z ∈ A → ∀x z ∈ A)
40		ax-17 925	. . . . . 6 ⊢ (z ∈ B → ∀x z ∈ B)
41	38, 39, 40	hbf 2751	. . . . 5 ⊢ ({⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–→B → ∀x{⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–→B)
42		feq1 2748	. . . . . 6 ⊢ (F = {⟨x, y⟩∣(x ∈ A ∧ y = C)} → (F:A–→B ↔ {⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–→B))
43	5, 42	ax-mp 6	. . . . 5 ⊢ (F:A–→B ↔ {⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–→B)
44	43	bial 695	. . . . 5 ⊢ (∀x F:A–→B ↔ ∀x{⟨x, y⟩∣(x ∈ A ∧ y = C)}:A–→B)
45	41, 43, 44	3imtr4 192	. . . 4 ⊢ (F:A–→B → ∀x F:A–→B)
46		ffvrn 2890	. . . . . . 7 ⊢ ((F:A–→B ∧ x ∈ A) → (F ‘x) ∈ B)
47	46	adantr 306	. . . . . 6 ⊢ (((F:A–→B ∧ x ∈ A) ∧ C ∈ V) → (F ‘x) ∈ B)
48		fvopab2 2878	. . . . . . . . 9 ⊢ ((x ∈ A ∧ C ∈ V) → ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x) = C)
49	5	fveq1i 2833	. . . . . . . . 9 ⊢ (F ‘x) = ({⟨x, y⟩∣(x ∈ A ∧ y = C)} ‘x)
50	48, 49	syl5eq 1136	. . . . . . . 8 ⊢ ((x ∈ A ∧ C ∈ V) → (F ‘x) = C)
51	50	eleq1d 1155	. . . . . . 7 ⊢ ((x ∈ A ∧ C ∈ V) → ((F ‘x) ∈ B ↔ C ∈ B))
52	51	adantll 309	. . . . . 6 ⊢ (((F:A–→B ∧ x ∈ A) ∧ C ∈ V) → ((F ‘x) ∈ B ↔ C ∈ B))
53	47, 52	mpbid 170	. . . . 5 ⊢ (((F:A–→B ∧ x ∈ A) ∧ C ∈ V) → C ∈ B)
54	53	exp 291	. . . 4 ⊢ ((F:A–→B ∧ x ∈ A) → (C ∈ V → C ∈ B))
55	45, 54	r19.20da 1255	. . 3 ⊢ (F:A–→B → (∀x ∈ A C ∈ V → ∀x ∈ A C ∈ B))
56	37, 55	mpd 46	. 2 ⊢ (F:A–→B → ∀x ∈ A C ∈ B)
57	29, 56	impbi 139	1 ⊢ (∀x ∈ A C ∈ B ↔ F:A–→B)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ⊆ wss 1487  {copab 2055  dom cdm 2410  ran crn 2411   Fn wfn 2417  –→wf 2418   ‘cfv 2422

This theorem is referenced by:  f1stres 3096  dom2d 3307  pw2en 3348  mapenlem2 3385  xpmapenlem4 3394  occllem4 5183  projlem24 5216  hosf 5602  hodf 5603  strlem3a 5693

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-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-fv 2438

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