Theorem tz6.12i-afv2 47980

Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6907. (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 2851	. . . . . . . . 9 ⊢ ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝑦 ∈ ran 𝐹))
2		dfatafv2rnb 47964	. . . . . . . . . . . 12 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3		dfdfat2 47865	. . . . . . . . . . . . 13 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
4	3	simprbi 502	. . . . . . . . . . . 12 ⊢ (𝐹 defAt 𝐴 → ∃!𝑦 𝐴𝐹𝑦)
5	2, 4	sylbir 238	. . . . . . . . . . 11 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
6		tz6.12c-afv2 47979	. . . . . . . . . . 11 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
7	5, 6	syl 18	. . . . . . . . . 10 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
8	7	biimpcd 252	. . . . . . . . 9 ⊢ ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹𝑦))
9	1, 8	sylbird 263	. . . . . . . 8 ⊢ ((𝐹''''𝐴) = 𝑦 → (𝑦 ∈ ran 𝐹 → 𝐴𝐹𝑦))
10	9	eqcoms 2771	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 → 𝐴𝐹𝑦))
11		eleq1 2851	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
12		breq2 5113	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝐴𝐹𝑦 ↔ 𝐴𝐹(𝐹''''𝐴)))
13	10, 11, 12	3imtr3d 296	. . . . . 6 ⊢ (𝑦 = (𝐹''''𝐴) → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴)))
14	13	vtocleg 3521	. . . . 5 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴)))
15	14	pm2.43i 53	. . . 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 2851	. . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝐵 ∈ ran 𝐹))
18		breq2 5113	. . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
19	16, 17, 18	3imtr3d 296	. 2 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐵 ∈ ran 𝐹 → 𝐴𝐹𝐵))
20	19	com12 33	1 ⊢ (𝐵 ∈ ran 𝐹 → ((𝐹''''𝐴) = 𝐵 → 𝐴𝐹𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∃!weu 2596   class class class wbr 5109  dom cdm 5661  ran crn 5662   defAt wdfat 47853  ''''cafv2 47945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-dfat 47856  df-afv2 47946

This theorem is referenced by: (None)