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Theorem tz6.12i-afv2 47255
Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6934. (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 2829 . . . . . . . . 9 ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹𝑦 ∈ ran 𝐹))
2 dfatafv2rnb 47239 . . . . . . . . . . . 12 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3 dfdfat2 47140 . . . . . . . . . . . . 13 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
43simprbi 496 . . . . . . . . . . . 12 (𝐹 defAt 𝐴 → ∃!𝑦 𝐴𝐹𝑦)
52, 4sylbir 235 . . . . . . . . . . 11 ((𝐹''''𝐴) ∈ ran 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
6 tz6.12c-afv2 47254 . . . . . . . . . . 11 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
75, 6syl 17 . . . . . . . . . 10 ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
87biimpcd 249 . . . . . . . . 9 ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹𝐴𝐹𝑦))
91, 8sylbird 260 . . . . . . . 8 ((𝐹''''𝐴) = 𝑦 → (𝑦 ∈ ran 𝐹𝐴𝐹𝑦))
109eqcoms 2745 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹𝐴𝐹𝑦))
11 eleq1 2829 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
12 breq2 5147 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴)))
1310, 11, 123imtr3d 293 . . . . . 6 (𝑦 = (𝐹''''𝐴) → ((𝐹''''𝐴) ∈ ran 𝐹𝐴𝐹(𝐹''''𝐴)))
1413vtocleg 3553 . . . . 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 2829 . . 3 ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹𝐵 ∈ ran 𝐹))
18 breq2 5147 . . 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 206   = wceq 1540  wcel 2108  ∃!weu 2568   class class class wbr 5143  dom cdm 5685  ran crn 5686   defAt wdfat 47128  ''''cafv2 47220
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-pow 5365  ax-pr 5432  ax-un 7755
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-nfc 2892  df-ne 2941  df-nel 3047  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-pw 4602  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-iota 6514  df-fun 6563  df-fn 6564  df-dfat 47131  df-afv2 47221
This theorem is referenced by: (None)
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