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Theorem fvimacnvALT 6811
 Description: Alternate proof of fvimacnv 6807, based on funimass3 6808. If funimass3 6808 is ever proved directly, as opposed to using funimacnv 6410 pointwise, then the proof of funimacnv 6410 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 4703 . . 3 (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
2 funimass3 6808 . . 3 ((Fun 𝐹 ∧ {𝐴} ⊆ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (𝐹𝐵)))
31, 2sylan2 595 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (𝐹𝐵)))
4 fvex 6665 . . . 4 (𝐹𝐴) ∈ V
54snss 4681 . . 3 ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵)
6 eqid 2798 . . . . . 6 dom 𝐹 = dom 𝐹
7 df-fn 6332 . . . . . . 7 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
87biimpri 231 . . . . . 6 ((Fun 𝐹 ∧ dom 𝐹 = dom 𝐹) → 𝐹 Fn dom 𝐹)
96, 8mpan2 690 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
10 fnsnfv 6725 . . . . 5 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
119, 10sylan 583 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
1211sseq1d 3947 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ({(𝐹𝐴)} ⊆ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵))
135, 12syl5bb 286 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵))
14 snssg 4680 . . 3 (𝐴 ∈ dom 𝐹 → (𝐴 ∈ (𝐹𝐵) ↔ {𝐴} ⊆ (𝐹𝐵)))
1514adantl 485 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝐵) ↔ {𝐴} ⊆ (𝐹𝐵)))
163, 13, 153bitr4d 314 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3882  {csn 4527  ◡ccnv 5521  dom cdm 5522   “ cima 5525  Fun wfun 6323   Fn wfn 6324  ‘cfv 6329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-iota 6288  df-fun 6331  df-fn 6332  df-fv 6337 This theorem is referenced by: (None)
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