Theorem fvopab2 3786

Description: Value of a function given by an ordered-pair class abstraction.

Assertion

Ref	Expression
fvopab2	⊢ ((x ∈ A ⋀ B ∈ C) → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = B)

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

Proof of Theorem fvopab2

Step	Hyp	Ref	Expression
1		elex 1816	. . 3 ⊢ (B ∈ C → ∃y y = B)
2		ax-17 970	. . . . 5 ⊢ (x ∈ A → ∀y x ∈ A)
3		hbopab2 2810	. . . . . . 7 ⊢ (z ∈ {⟨x, y⟩∣(x ∈ A ⋀ y = B)} → ∀y z ∈ {⟨x, y⟩∣(x ∈ A ⋀ y = B)})
4		ax-17 970	. . . . . . 7 ⊢ (z ∈ x → ∀y z ∈ x)
5	3, 4	hbfv 3724	. . . . . 6 ⊢ (z ∈ ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) → ∀y z ∈ ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x))
6		ax-17 970	. . . . . 6 ⊢ (z ∈ B → ∀y z ∈ B)
7	5, 6	hbeq 1563	. . . . 5 ⊢ (({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = B → ∀y({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = B)
8	2, 7	hbim 1006	. . . 4 ⊢ ((x ∈ A → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = B) → ∀y(x ∈ A → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = B))
9		visset 1810	. . . . . . . 8 ⊢ y ∈ V
10		eleq1 1532	. . . . . . . 8 ⊢ (y = B → (y ∈ V ↔ B ∈ V))
11	9, 10	mpbii 193	. . . . . . 7 ⊢ (y = B → B ∈ V)
12	3	tz6.12f 3733	. . . . . . . . . . 11 ⊢ ((⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ A ⋀ y = B)} ⋀ ∃!y⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ A ⋀ y = B)}) → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = y)
13		opabid 2806	. . . . . . . . . . 11 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ A ⋀ y = B)} ↔ (x ∈ A ⋀ y = B))
14	12, 13	sylanbr 450	. . . . . . . . . 10 ⊢ (((x ∈ A ⋀ y = B) ⋀ ∃!y⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ A ⋀ y = B)}) → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = y)
15		euanv 1431	. . . . . . . . . . 11 ⊢ (∃!y(x ∈ A ⋀ y = B) ↔ (x ∈ A ⋀ ∃!y y = B))
16	13	eubii 1386	. . . . . . . . . . 11 ⊢ (∃!y⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ A ⋀ y = B)} ↔ ∃!y(x ∈ A ⋀ y = B))
17		eueq 1913	. . . . . . . . . . . 12 ⊢ (B ∈ V ↔ ∃!y y = B)
18	17	anbi2i 480	. . . . . . . . . . 11 ⊢ ((x ∈ A ⋀ B ∈ V) ↔ (x ∈ A ⋀ ∃!y y = B))
19	15, 16, 18	3bitr4r 184	. . . . . . . . . 10 ⊢ ((x ∈ A ⋀ B ∈ V) ↔ ∃!y⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ A ⋀ y = B)})
20	14, 19	sylan2b 452	. . . . . . . . 9 ⊢ (((x ∈ A ⋀ y = B) ⋀ (x ∈ A ⋀ B ∈ V)) → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = y)
21	20	exp43 384	. . . . . . . 8 ⊢ (x ∈ A → (y = B → (x ∈ A → (B ∈ V → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = y))))
22	21	pm2.43a 66	. . . . . . 7 ⊢ (x ∈ A → (y = B → (B ∈ V → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = y)))
23	11, 22	mpdi 48	. . . . . 6 ⊢ (x ∈ A → (y = B → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = y))
24	23	com12 11	. . . . 5 ⊢ (y = B → (x ∈ A → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = y))
25		eqeq2 1482	. . . . 5 ⊢ (y = B → (({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = y ↔ ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = B))
26	24, 25	sylibd 202	. . . 4 ⊢ (y = B → (x ∈ A → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = B))
27	8, 26	19.23ai 1063	. . 3 ⊢ (∃y y = B → (x ∈ A → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = B))
28	1, 27	syl 10	. 2 ⊢ (B ∈ C → (x ∈ A → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = B))
29	28	impcom 351	1 ⊢ ((x ∈ A ⋀ B ∈ C) → ({⟨x, y⟩∣(x ∈ A ⋀ y = B)} ‘x) = B)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 955   ∈ wcel 957  ∃wex 979  ∃!weu 1379  Vcvv 1808  ⟨cop 2408  {copab 2662   ‘cfv 3178

This theorem is referenced by:  elrnopabg 3795  fopab2 3818  rnssopab 3820  fopabco 3827  fopabsn 3835  abrexexlem2 3854  dom2d 4394  pw2en 4435  mapxpen 4484  xpmapenlem2 4486  xpmapenlem4 4488  xpmapenlem5 4489  ac6lem 4737  iundom 4795  sumeqfv 6950  serzfsum 6957  fsum1 6958  fsump1 6959  isumval2t 7147  isumclim3t 7152  isumcmpi 7167  geoisum1c 7197  fsumcnlem 7951  cnlnadjlem5 9960  fvopab2a 10417

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-cnv 3182  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fv 3194

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