Theorem funopfv 2886

Description: Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41.

Assertion

Ref	Expression
funopfv	⊢ ((Fun F ∧ A ∈ dom F) → ⟨A, (F ‘A)⟩ ∈ F)

Proof of Theorem funopfv

Step	Hyp	Ref	Expression
1		fvex 2838	. . 3 ⊢ (F ‘A) ∈ V
2	1	isseti 1352	. 2 ⊢ ∃x x = (F ‘A)
3		visset 1350	. . . . . . 7 ⊢ x ∈ V
4	3	funfvop 2857	. . . . . 6 ⊢ ((Fun F ∧ A ∈ dom F) → ((F ‘A) = x ↔ ⟨A, x⟩ ∈ F))
5		opeq2 1877	. . . . . . . 8 ⊢ ((F ‘A) = x → ⟨A, (F ‘A)⟩ = ⟨A, x⟩)
6	5	eleq1d 1155	. . . . . . 7 ⊢ ((F ‘A) = x → (⟨A, (F ‘A)⟩ ∈ F ↔ ⟨A, x⟩ ∈ F))
7	6	biimprcd 138	. . . . . 6 ⊢ (⟨A, x⟩ ∈ F → ((F ‘A) = x → ⟨A, (F ‘A)⟩ ∈ F))
8	4, 7	syl6bi 187	. . . . 5 ⊢ ((Fun F ∧ A ∈ dom F) → ((F ‘A) = x → ((F ‘A) = x → ⟨A, (F ‘A)⟩ ∈ F)))
9	8	pm2.43d 59	. . . 4 ⊢ ((Fun F ∧ A ∈ dom F) → ((F ‘A) = x → ⟨A, (F ‘A)⟩ ∈ F))
10		cleqcom 1103	. . . 4 ⊢ (x = (F ‘A) ↔ (F ‘A) = x)
11	9, 10	syl5ib 181	. . 3 ⊢ ((Fun F ∧ A ∈ dom F) → (x = (F ‘A) → ⟨A, (F ‘A)⟩ ∈ F))
12	11	19.23adv 954	. 2 ⊢ ((Fun F ∧ A ∈ dom F) → (∃x x = (F ‘A) → ⟨A, (F ‘A)⟩ ∈ F))
13	2, 12	mpi 44	1 ⊢ ((Fun F ∧ A ∈ dom F) → ⟨A, (F ‘A)⟩ ∈ F)

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  dom cdm 2410  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  fnopfv 2887  fvrn 2888  funfvima3 2906  fundmen 3333

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-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-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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