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Theorem tz6.12i-afv2 43592
 Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6669. (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 2899 . . . . . . . . 9 ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹𝑦 ∈ ran 𝐹))
2 dfatafv2rnb 43576 . . . . . . . . . . . 12 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3 dfdfat2 43477 . . . . . . . . . . . . 13 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
43simprbi 500 . . . . . . . . . . . 12 (𝐹 defAt 𝐴 → ∃!𝑦 𝐴𝐹𝑦)
52, 4sylbir 238 . . . . . . . . . . 11 ((𝐹''''𝐴) ∈ ran 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
6 tz6.12c-afv2 43591 . . . . . . . . . . 11 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
75, 6syl 17 . . . . . . . . . 10 ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
87biimpcd 252 . . . . . . . . 9 ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹𝐴𝐹𝑦))
91, 8sylbird 263 . . . . . . . 8 ((𝐹''''𝐴) = 𝑦 → (𝑦 ∈ ran 𝐹𝐴𝐹𝑦))
109eqcoms 2829 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹𝐴𝐹𝑦))
11 eleq1 2899 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
12 breq2 5043 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴)))
1310, 11, 123imtr3d 296 . . . . . 6 (𝑦 = (𝐹''''𝐴) → ((𝐹''''𝐴) ∈ ran 𝐹𝐴𝐹(𝐹''''𝐴)))
1413vtocleg 3558 . . . . 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 2899 . . 3 ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹𝐵 ∈ ran 𝐹))
18 breq2 5043 . . 3 ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
1916, 17, 183imtr3d 296 . 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 209   = wceq 1538   ∈ wcel 2115  ∃!weu 2653   class class class wbr 5039  dom cdm 5528  ran crn 5529   defAt wdfat 43465  ''''cafv2 43557 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-nel 3112  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-iota 6287  df-fun 6330  df-fn 6331  df-dfat 43468  df-afv2 43558 This theorem is referenced by: (None)
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