Theorem fvimacnv 7001

Description: The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 6575 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
fvimacnv	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))

Proof of Theorem fvimacnv

Step	Hyp	Ref	Expression
1		funfvop 6998	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
2		fvex 6847	. . . . . . 7 ⊢ (𝐹‘𝐴) ∈ V
3		opelcnvg 5829	. . . . . . 7 ⊢ (((𝐹‘𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (⟨(𝐹‘𝐴), 𝐴⟩ ∈ ◡𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
4	2, 3	mpan 696	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 → (⟨(𝐹‘𝐴), 𝐴⟩ ∈ ◡𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
5	4	adantl 482	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (⟨(𝐹‘𝐴), 𝐴⟩ ∈ ◡𝐹 ↔ ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹))
6	1, 5	mpbird 258	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨(𝐹‘𝐴), 𝐴⟩ ∈ ◡𝐹)
7		elimasng 6048	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) ↔ ⟨(𝐹‘𝐴), 𝐴⟩ ∈ ◡𝐹))
8	2, 7	mpan 696	. . . . 5 ⊢ (𝐴 ∈ dom 𝐹 → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) ↔ ⟨(𝐹‘𝐴), 𝐴⟩ ∈ ◡𝐹))
9	8	adantl 482	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) ↔ ⟨(𝐹‘𝐴), 𝐴⟩ ∈ ◡𝐹))
10	6, 9	mpbird 258	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}))
11	2	snss 4723	. . . . 5 ⊢ ((𝐹‘𝐴) ∈ 𝐵 ↔ {(𝐹‘𝐴)} ⊆ 𝐵)
12		imass2 6061	. . . . 5 ⊢ ({(𝐹‘𝐴)} ⊆ 𝐵 → (◡𝐹 “ {(𝐹‘𝐴)}) ⊆ (◡𝐹 “ 𝐵))
13	11, 12	sylbi 218	. . . 4 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → (◡𝐹 “ {(𝐹‘𝐴)}) ⊆ (◡𝐹 “ 𝐵))
14	13	sseld 3921	. . 3 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → (𝐴 ∈ (◡𝐹 “ {(𝐹‘𝐴)}) → 𝐴 ∈ (◡𝐹 “ 𝐵)))
15	10, 14	syl5com 31	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 → 𝐴 ∈ (◡𝐹 “ 𝐵)))
16		fvimacnvi 7000	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵)
17	16	ex 413	. . 3 ⊢ (Fun 𝐹 → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵))
18	17	adantr 481	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ 𝐵) → (𝐹‘𝐴) ∈ 𝐵))
19	15, 18	impbid 213	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  Vcvv 3432   ⊆ wss 3890  {csn 4562  ⟨cop 4568  ^◡ccnv 5624  dom cdm 5625   “ cima 5628  Fun wfun 6486  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  funimass3  7002  elpreima  7006  iinpreima  7017  rhmpreimaidl  21277  isr0  23727  rnelfmlem  23942  rnelfm  23943  fmfnfmlem2  23945  fmfnfmlem4  23947  fmfnfm  23948  metustid  24544  metustsym  24545  metustexhalf  24546  xppreima  32744  dstfrvel  34665  ballotlemrv  34711  bj-fvimacnv0  37653  bj-isrvec  37661  grpokerinj  38267  diaintclN  41557  dibintclN  41666  dihintcl  41843  aks6d1c2lem4  42619  aks6d1c6lem2  42663  rhmqusspan  42677  arearect  43667  areaquad  43668  tannpoly  47360  sinnpoly  47361