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Theorem fvimacnvALT 6302
Description: Alternate proof of fvimacnv 6298, based on funimass3 6299. If funimass3 6299 is ever proved directly, as opposed to using funimacnv 5938 pointwise, then the proof of funimacnv 5938 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 4315 . . 3 (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
2 funimass3 6299 . . 3 ((Fun 𝐹 ∧ {𝐴} ⊆ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (𝐹𝐵)))
31, 2sylan2 491 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (𝐹𝐵)))
4 fvex 6168 . . . 4 (𝐹𝐴) ∈ V
54snss 4293 . . 3 ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵)
6 eqid 2621 . . . . . 6 dom 𝐹 = dom 𝐹
7 df-fn 5860 . . . . . . 7 (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
87biimpri 218 . . . . . 6 ((Fun 𝐹 ∧ dom 𝐹 = dom 𝐹) → 𝐹 Fn dom 𝐹)
96, 8mpan2 706 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
10 fnsnfv 6225 . . . . 5 ((𝐹 Fn dom 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
119, 10sylan 488 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → {(𝐹𝐴)} = (𝐹 “ {𝐴}))
1211sseq1d 3617 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ({(𝐹𝐴)} ⊆ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵))
135, 12syl5bb 272 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵))
14 snssg 4303 . . 3 (𝐴 ∈ dom 𝐹 → (𝐴 ∈ (𝐹𝐵) ↔ {𝐴} ⊆ (𝐹𝐵)))
1514adantl 482 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝐵) ↔ {𝐴} ⊆ (𝐹𝐵)))
163, 13, 153bitr4d 300 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3560  {csn 4155  ccnv 5083  dom cdm 5084  cima 5087  Fun wfun 5851   Fn wfn 5852  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865
This theorem is referenced by: (None)
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