Theorem tz6.12i-afv2 46536

Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. analogous to tz6.12i 6919. (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 2816	. . . . . . . . 9 ⊢ ((𝐹''''𝐴) = 𝑦 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝑦 ∈ ran 𝐹))
2		dfatafv2rnb 46520	. . . . . . . . . . . 12 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3		dfdfat2 46421	. . . . . . . . . . . . 13 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))
4	3	simprbi 496	. . . . . . . . . . . 12 ⊢ (𝐹 defAt 𝐴 → ∃!𝑦 𝐴𝐹𝑦)
5	2, 4	sylbir 234	. . . . . . . . . . 11 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → ∃!𝑦 𝐴𝐹𝑦)
6		tz6.12c-afv2 46535	. . . . . . . . . . 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 260	. . . . . . . 8 ⊢ ((𝐹''''𝐴) = 𝑦 → (𝑦 ∈ ran 𝐹 → 𝐴𝐹𝑦))
10	9	eqcoms 2735	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 → 𝐴𝐹𝑦))
11		eleq1 2816	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝑦 ∈ ran 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹))
12		breq2 5146	. . . . . . 7 ⊢ (𝑦 = (𝐹''''𝐴) → (𝐴𝐹𝑦 ↔ 𝐴𝐹(𝐹''''𝐴)))
13	10, 11, 12	3imtr3d 293	. . . . . 6 ⊢ (𝑦 = (𝐹''''𝐴) → ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐴𝐹(𝐹''''𝐴)))
14	13	vtocleg 3537	. . . . 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 2816	. . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝐵 ∈ ran 𝐹))
18		breq2 5146	. . 3 ⊢ ((𝐹''''𝐴) = 𝐵 → (𝐴𝐹(𝐹''''𝐴) ↔ 𝐴𝐹𝐵))
19	16, 17, 18	3imtr3d 293	. 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 1534   ∈ wcel 2099  ∃!weu 2557   class class class wbr 5142  dom cdm 5672  ran crn 5673   defAt wdfat 46409  ''''cafv2 46501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-dfat 46412  df-afv2 46502

This theorem is referenced by: (None)