Theorem fvelima 2859

Description: Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42.

Assertion

Ref	Expression
fvelima	⊢ ((Fun F ∧ A ∈ (F “ B)) → ∃x ∈ B (F ‘x) = A)

Distinct variable group(s):   x,A   x,B   x,F

Proof of Theorem fvelima

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . . 6 ⊢ (y = A → (y ∈ (F “ B) ↔ A ∈ (F “ B)))
2		cleq2 1110	. . . . . . 7 ⊢ (y = A → ((F ‘x) = y ↔ (F ‘x) = A))
3	2	birexdv 1220	. . . . . 6 ⊢ (y = A → (∃x ∈ B (F ‘x) = y ↔ ∃x ∈ B (F ‘x) = A))
4	1, 3	imbi12d 474	. . . . 5 ⊢ (y = A → ((y ∈ (F “ B) → ∃x ∈ B (F ‘x) = y) ↔ (A ∈ (F “ B) → ∃x ∈ B (F ‘x) = A)))
5	4	imbi2d 464	. . . 4 ⊢ (y = A → ((Fun F → (y ∈ (F “ B) → ∃x ∈ B (F ‘x) = y)) ↔ (Fun F → (A ∈ (F “ B) → ∃x ∈ B (F ‘x) = A))))
6		visset 1350	. . . . . . . 8 ⊢ y ∈ V
7	6	funfvopi 2853	. . . . . . 7 ⊢ (Fun F → (⟨x, y⟩ ∈ F → (F ‘x) = y))
8		df-br 2063	. . . . . . 7 ⊢ (xFy ↔ ⟨x, y⟩ ∈ F)
9	7, 8	syl5ib 181	. . . . . 6 ⊢ (Fun F → (xFy → (F ‘x) = y))
10	9	r19.22sdv 1279	. . . . 5 ⊢ (Fun F → (∃x ∈ B xFy → ∃x ∈ B (F ‘x) = y))
11	6	elima 2606	. . . . 5 ⊢ (y ∈ (F “ B) ↔ ∃x ∈ B xFy)
12	10, 11	syl5ib 181	. . . 4 ⊢ (Fun F → (y ∈ (F “ B) → ∃x ∈ B (F ‘x) = y))
13	5, 12	vtoclg 1383	. . 3 ⊢ (A ∈ (F “ B) → (Fun F → (A ∈ (F “ B) → ∃x ∈ B (F ‘x) = A)))
14	13	pm2.43b 61	. 2 ⊢ (Fun F → (A ∈ (F “ B) → ∃x ∈ B (F ‘x) = A))
15	14	imp 277	1 ⊢ ((Fun F ∧ A ∈ (F “ B)) → ∃x ∈ B (F ‘x) = A)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ⟨cop 1810   class class class wbr 2054   “ cima 2413  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  isofrlem 2939  tz7.49 2997  zornlem5 3607  zornlem6 3608

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-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-fv 2438

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