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Theorem fvimacnvALT 7077
Description: Alternate proof of fvimacnv 7073, based on funimass3 7074. If funimass3 7074 is ever proved directly, as opposed to using funimacnv 6647 pointwise, then the proof of funimacnv 6647 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fvimacnvALT ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))

Proof of Theorem fvimacnvALT
StepHypRef Expression
1 snssi 4808 . . 3 (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
2 funimass3 7074 . . 3 ((Fun 𝐹 ∧ {𝐴} ⊆ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (𝐹𝐵)))
31, 2sylan2 593 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (𝐹𝐵)))
4 fvex 6919 . . . 4 (𝐹𝐴) ∈ V
54snss 4785 . . 3 ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵)
6 eqid 2737 . . . . . 6 dom 𝐹 = dom 𝐹
7 df-fn 6564 . . . . . . 7 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
87biimpri 228 . . . . . 6 ((Fun 𝐹 ∧ dom 𝐹 = dom 𝐹) → 𝐹 Fn dom 𝐹)
96, 8mpan2 691 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
10 fnsnfv 6988 . . . . 5 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
119, 10sylan 580 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
1211sseq1d 4015 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ({(𝐹𝐴)} ⊆ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵))
135, 12bitrid 283 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵))
14 snssg 4783 . . 3 (𝐴 ∈ dom 𝐹 → (𝐴 ∈ (𝐹𝐵) ↔ {𝐴} ⊆ (𝐹𝐵)))
1514adantl 481 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝐵) ↔ {𝐴} ⊆ (𝐹𝐵)))
163, 13, 153bitr4d 311 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  {csn 4626  ccnv 5684  dom cdm 5685  cima 5688  Fun wfun 6555   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by: (None)
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