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Theorem fvimacnv 5203
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 4899 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
fvimacnv ((Fun 𝐹 A dom 𝐹) → ((𝐹A) BA (𝐹B)))

Proof of Theorem fvimacnv
StepHypRef Expression
1 funfvop 5200 . . . . 5 ((Fun 𝐹 A dom 𝐹) → ⟨A, (𝐹A)⟩ 𝐹)
2 funfvex 5113 . . . . . 6 ((Fun 𝐹 A dom 𝐹) → (𝐹A) V)
3 opelcnvg 4438 . . . . . 6 (((𝐹A) V A dom 𝐹) → (⟨(𝐹A), A 𝐹 ↔ ⟨A, (𝐹A)⟩ 𝐹))
42, 3sylancom 399 . . . . 5 ((Fun 𝐹 A dom 𝐹) → (⟨(𝐹A), A 𝐹 ↔ ⟨A, (𝐹A)⟩ 𝐹))
51, 4mpbird 156 . . . 4 ((Fun 𝐹 A dom 𝐹) → ⟨(𝐹A), A 𝐹)
6 elimasng 4616 . . . . 5 (((𝐹A) V A dom 𝐹) → (A (𝐹 “ {(𝐹A)}) ↔ ⟨(𝐹A), A 𝐹))
72, 6sylancom 399 . . . 4 ((Fun 𝐹 A dom 𝐹) → (A (𝐹 “ {(𝐹A)}) ↔ ⟨(𝐹A), A 𝐹))
85, 7mpbird 156 . . 3 ((Fun 𝐹 A dom 𝐹) → A (𝐹 “ {(𝐹A)}))
9 snssg 3470 . . . . . . . 8 ((𝐹A) V → ((𝐹A) B ↔ {(𝐹A)} ⊆ B))
102, 9syl 14 . . . . . . 7 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) B ↔ {(𝐹A)} ⊆ B))
11 imass2 4624 . . . . . . 7 ({(𝐹A)} ⊆ B → (𝐹 “ {(𝐹A)}) ⊆ (𝐹B))
1210, 11syl6bi 152 . . . . . 6 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) B → (𝐹 “ {(𝐹A)}) ⊆ (𝐹B)))
1312imp 115 . . . . 5 (((Fun 𝐹 A dom 𝐹) (𝐹A) B) → (𝐹 “ {(𝐹A)}) ⊆ (𝐹B))
1413sseld 2917 . . . 4 (((Fun 𝐹 A dom 𝐹) (𝐹A) B) → (A (𝐹 “ {(𝐹A)}) → A (𝐹B)))
1514ex 108 . . 3 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) B → (A (𝐹 “ {(𝐹A)}) → A (𝐹B))))
168, 15mpid 37 . 2 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) BA (𝐹B)))
17 fvimacnvi 5202 . . . 4 ((Fun 𝐹 A (𝐹B)) → (𝐹A) B)
1817ex 108 . . 3 (Fun 𝐹 → (A (𝐹B) → (𝐹A) B))
1918adantr 261 . 2 ((Fun 𝐹 A dom 𝐹) → (A (𝐹B) → (𝐹A) B))
2016, 19impbid 120 1 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) BA (𝐹B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1370  Vcvv 2531  wss 2890  {csn 3346  cop 3349  ccnv 4267  dom cdm 4268  cima 4271  Fun wfun 4819  cfv 4825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833
This theorem is referenced by:  funimass3  5204  elpreima  5207
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