Theorem tz6.12i-afv2 48449

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 2848	. . . . . . . . 9 ⊢ ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝑦 ∈ ran 𝐹))
2		dfatafv2rnb 48433	. . . . . . . . . . . 12 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3		dfdfat2 48334	. . . . . . . . . . . . 13 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
4	3	simprbi 503	. . . . . . . . . . . 12 ⊢ (𝐹 defAt 𝐴 → ∃!𝑦 𝐴𝐹𝑦)
5	2, 4	sylbir 238	. . . . . . . . . . 11 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
6		tz6.12c-afv2 48448	. . . . . . . . . . 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 2768	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 → 𝐴𝐹𝑦))
11		eleq1 2848	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
12		breq2 5106	. . . . . . 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 3516	. . . . 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 2848	. . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝐵 ∈ ran 𝐹))
18		breq2 5106	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144  ∃!weu 2593   class class class wbr 5102  dom cdm 5651  ran crn 5652   defAt wdfat 48323  ''''cafv2 48414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6497  df-fun 6545  df-fn 6546  df-dfat 48326  df-afv2 48415

This theorem is used by: (None)