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Theorem tz6.12i-afv2 47193
Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6935. (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 2827 . . . . . . . . 9 ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹𝑦 ∈ ran 𝐹))
2 dfatafv2rnb 47177 . . . . . . . . . . . 12 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3 dfdfat2 47078 . . . . . . . . . . . . 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 47192 . . . . . . . . . . 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 2743 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹𝐴𝐹𝑦))
11 eleq1 2827 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
12 breq2 5152 . . . . . . 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 2827 . . 3 ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹𝐵 ∈ ran 𝐹))
18 breq2 5152 . . 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 1537  wcel 2106  ∃!weu 2566   class class class wbr 5148  dom cdm 5689  ran crn 5690   defAt wdfat 47066  ''''cafv2 47158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-dfat 47069  df-afv2 47159
This theorem is referenced by: (None)
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