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Theorem tz6.12i-afv2 44735
Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6800. (Contributed by AV, 5-Sep-2022.)
Assertion
Ref Expression
tz6.12i-afv2 (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))

Proof of Theorem tz6.12i-afv2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2826 . . . . . . . . 9 ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹𝑦 ∈ ran 𝐹))
2 dfatafv2rnb 44719 . . . . . . . . . . . 12 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3 dfdfat2 44620 . . . . . . . . . . . . 13 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
43simprbi 497 . . . . . . . . . . . 12 (𝐹 defAt 𝐴 → ∃!𝑦 𝐴𝐹𝑦)
52, 4sylbir 234 . . . . . . . . . . 11 ((𝐹''''𝐴) ∈ ran 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
6 tz6.12c-afv2 44734 . . . . . . . . . . 11 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
75, 6syl 17 . . . . . . . . . 10 ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
87biimpcd 248 . . . . . . . . 9 ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹𝐴𝐹𝑦))
91, 8sylbird 259 . . . . . . . 8 ((𝐹''''𝐴) = 𝑦 → (𝑦 ∈ ran 𝐹𝐴𝐹𝑦))
109eqcoms 2746 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹𝐴𝐹𝑦))
11 eleq1 2826 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
12 breq2 5078 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴)))
1310, 11, 123imtr3d 293 . . . . . 6 (𝑦 = (𝐹''''𝐴) → ((𝐹''''𝐴) ∈ ran 𝐹𝐴𝐹(𝐹''''𝐴)))
1413vtocleg 3521 . . . . 5 ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹𝐴𝐹(𝐹''''𝐴)))
1514pm2.43i 52 . . . 4 ((𝐹''''𝐴) ∈ ran 𝐹𝐴𝐹(𝐹''''𝐴))
1615a1i 11 . . 3 ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹𝐴𝐹(𝐹''''𝐴)))
17 eleq1 2826 . . 3 ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹𝐵 ∈ ran 𝐹))
18 breq2 5078 . . 3 ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
1916, 17, 183imtr3d 293 . 2 ((𝐹''''𝐴) = 𝐵 → (𝐵 ∈ ran 𝐹𝐴𝐹𝐵))
2019com12 32 1 (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2106  ∃!weu 2568   class class class wbr 5074  dom cdm 5589  ran crn 5590   defAt wdfat 44608  ''''cafv2 44700
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-dfat 44611  df-afv2 44701
This theorem is referenced by: (None)
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