Theorem tz6.12i-afv2 45937

Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6916. (Contributed by AV, 5-Sep-2022.)

Assertion

Ref	Expression
tz6.12i-afv2	⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵))

Proof of Theorem tz6.12i-afv2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2821	. . . . . . . . 9 ⊢ ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝑦 ∈ ran 𝐹))
2		dfatafv2rnb 45921	. . . . . . . . . . . 12 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3		dfdfat2 45822	. . . . . . . . . . . . 13 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
4	3	simprbi 497	. . . . . . . . . . . 12 ⊢ (𝐹 defAt 𝐴 → ∃!𝑦 𝐴𝐹𝑦)
5	2, 4	sylbir 234	. . . . . . . . . . 11 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
6		tz6.12c-afv2 45936	. . . . . . . . . . 11 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
8	7	biimpcd 248	. . . . . . . . 9 ⊢ ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹𝑦))
9	1, 8	sylbird 259	. . . . . . . 8 ⊢ ((𝐹''''𝐴) = 𝑦 → (𝑦 ∈ ran 𝐹 → 𝐴𝐹𝑦))
10	9	eqcoms 2740	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 → 𝐴𝐹𝑦))
11		eleq1 2821	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
12		breq2 5151	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝐴𝐹𝑦 ↔ 𝐴𝐹(𝐹''''𝐴)))
13	10, 11, 12	3imtr3d 292	. . . . . 6 ⊢ (𝑦 = (𝐹''''𝐴) → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴)))
14	13	vtocleg 3545	. . . . 5 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴)))
15	14	pm2.43i 52	. . . 4 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴))
16	15	a1i 11	. . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴)))
17		eleq1 2821	. . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝐵 ∈ ran 𝐹))
18		breq2 5151	. . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
19	16, 17, 18	3imtr3d 292	. 2 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐵 ∈ ran 𝐹 → 𝐴𝐹𝐵))
20	19	com12 32	1 ⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∃!weu 2562   class class class wbr 5147  dom cdm 5675  ran crn 5676   defAt wdfat 45810  ''''cafv2 45902

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fn 6543  df-dfat 45813  df-afv2 45903

This theorem is referenced by: (None)