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Theorem tz6.12i-afv2 46682
Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6918. (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 2813 . . . . . . . . 9 ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹𝑦 ∈ ran 𝐹))
2 dfatafv2rnb 46666 . . . . . . . . . . . 12 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3 dfdfat2 46567 . . . . . . . . . . . . 13 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
43simprbi 495 . . . . . . . . . . . 12 (𝐹 defAt 𝐴 → ∃!𝑦 𝐴𝐹𝑦)
52, 4sylbir 234 . . . . . . . . . . 11 ((𝐹''''𝐴) ∈ ran 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
6 tz6.12c-afv2 46681 . . . . . . . . . . 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 2733 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹𝐴𝐹𝑦))
11 eleq1 2813 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
12 breq2 5148 . . . . . . 7 (𝑦 = (𝐹''''𝐴) → (𝐴𝐹𝑦𝐴𝐹(𝐹''''𝐴)))
1310, 11, 123imtr3d 292 . . . . . 6 (𝑦 = (𝐹''''𝐴) → ((𝐹''''𝐴) ∈ ran 𝐹𝐴𝐹(𝐹''''𝐴)))
1413vtocleg 3532 . . . . 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 2813 . . 3 ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹𝐵 ∈ ran 𝐹))
18 breq2 5148 . . 3 ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
1916, 17, 183imtr3d 292 . 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 1533  wcel 2098  ∃!weu 2556   class class class wbr 5144  dom cdm 5673  ran crn 5674   defAt wdfat 46555  ''''cafv2 46647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-dfat 46558  df-afv2 46648
This theorem is referenced by: (None)
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