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Theorem fvimacnvALT 7053
Description: Alternate proof of fvimacnv 7049, based on funimass3 7050. If funimass3 7050 is ever proved directly, as opposed to using funimacnv 6625 pointwise, then the proof of funimacnv 6625 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 4809 . . 3 (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
2 funimass3 7050 . . 3 ((Fun 𝐹 ∧ {𝐴} ⊆ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (𝐹𝐵)))
31, 2sylan2 594 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (𝐹𝐵)))
4 fvex 6900 . . . 4 (𝐹𝐴) ∈ V
54snss 4787 . . 3 ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵)
6 eqid 2733 . . . . . 6 dom 𝐹 = dom 𝐹
7 df-fn 6542 . . . . . . 7 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
87biimpri 227 . . . . . 6 ((Fun 𝐹 ∧ dom 𝐹 = dom 𝐹) → 𝐹 Fn dom 𝐹)
96, 8mpan2 690 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
10 fnsnfv 6965 . . . . 5 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
119, 10sylan 581 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
1211sseq1d 4011 . . 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 4785 . . 3 (𝐴 ∈ dom 𝐹 → (𝐴 ∈ (𝐹𝐵) ↔ {𝐴} ⊆ (𝐹𝐵)))
1514adantl 483 . 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 205  wa 397   = wceq 1542  wcel 2107  wss 3946  {csn 4626  ccnv 5673  dom cdm 5674  cima 5677  Fun wfun 6533   Fn wfn 6534  cfv 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-fv 6547
This theorem is referenced by: (None)
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