Theorem funfvop 3798

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

Assertion

Ref	Expression
funfvop	⊢ ((Fun F ⋀ A ∈ dom F) → ⟨A, (F ‘A)⟩ ∈ F)

Proof of Theorem funfvop

Step	Hyp	Ref	Expression
1		fvex 3727	. . 3 ⊢ (F ‘A) ∈ V
2	1	isseti 1812	. 2 ⊢ ∃x x = (F ‘A)
3		visset 1810	. . . . . . 7 ⊢ x ∈ V
4	3	funopfvb 3751	. . . . . 6 ⊢ ((Fun F ⋀ A ∈ dom F) → ((F ‘A) = x ↔ ⟨A, x⟩ ∈ F))
5		opeq2 2485	. . . . . . . 8 ⊢ ((F ‘A) = x → ⟨A, (F ‘A)⟩ = ⟨A, x⟩)
6	5	eleq1d 1538	. . . . . . 7 ⊢ ((F ‘A) = x → (⟨A, (F ‘A)⟩ ∈ F ↔ ⟨A, x⟩ ∈ F))
7	6	biimprcd 156	. . . . . 6 ⊢ (⟨A, x⟩ ∈ F → ((F ‘A) = x → ⟨A, (F ‘A)⟩ ∈ F))
8	4, 7	syl6bi 214	. . . . 5 ⊢ ((Fun F ⋀ A ∈ dom F) → ((F ‘A) = x → ((F ‘A) = x → ⟨A, (F ‘A)⟩ ∈ F)))
9	8	pm2.43d 65	. . . 4 ⊢ ((Fun F ⋀ A ∈ dom F) → ((F ‘A) = x → ⟨A, (F ‘A)⟩ ∈ F))
10		eqcom 1475	. . . 4 ⊢ (x = (F ‘A) ↔ (F ‘A) = x)
11	9, 10	syl5ib 206	. . 3 ⊢ ((Fun F ⋀ A ∈ dom F) → (x = (F ‘A) → ⟨A, (F ‘A)⟩ ∈ F))
12	11	19.23adv 1213	. 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 3   ⋀ wa 223   = wceq 955   ∈ wcel 957  ∃wex 979  ⟨cop 2408  dom cdm 3166  Fun wfun 3172   ‘cfv 3178

This theorem is referenced by:  fvimacnv 3800  fnopfv 3806  fvelrn 3807  dff2 3812  funfvima3 3849  fundmen 4418  adjt 9814

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-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  ax-un 2862

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-id 2831  df-xp 3180  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194

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