Theorem imaeqsexvOLD 7355

Description: Obsolete version of rexima 7229 as of 14-Aug-2025. Duplicate version of rexima 7229. (Contributed by Scott Fenton, 27-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
imaeqsexvOLD.1	⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
imaeqsexvOLD	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)

Proof of Theorem imaeqsexvOLD

Step	Hyp	Ref	Expression
1		df-rex 3061	. . 3 ⊢ (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐹 “ 𝐵) ∧ 𝜑))
2		fvelimab 6950	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
3	2	anbi1d 631	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ 𝐵) ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑)))
4	3	exbidv 1921	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥(𝑥 ∈ (𝐹 “ 𝐵) ∧ 𝜑) ↔ ∃𝑥(∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑)))
5	1, 4	bitrid 283	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑥(∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑)))
6		rexcom4 3269	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑥∃𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑥 ∧ 𝜑))
7		eqcom 2742	. . . . . . 7 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
8	7	anbi1i 624	. . . . . 6 ⊢ (((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ (𝑥 = (𝐹‘𝑦) ∧ 𝜑))
9	8	exbii 1848	. . . . 5 ⊢ (∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝑥 = (𝐹‘𝑦) ∧ 𝜑))
10		fvex 6888	. . . . . 6 ⊢ (𝐹‘𝑦) ∈ V
11		imaeqsexvOLD.1	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
12	10, 11	ceqsexv 3511	. . . . 5 ⊢ (∃𝑥(𝑥 = (𝐹‘𝑦) ∧ 𝜑) ↔ 𝜓)
13	9, 12	bitri 275	. . . 4 ⊢ (∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ 𝜓)
14	13	rexbii 3083	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓)
15		r19.41v 3174	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑))
16	15	exbii 1848	. . 3 ⊢ (∃𝑥∃𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑥(∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑))
17	6, 14, 16	3bitr3ri 302	. 2 ⊢ (∃𝑥(∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓)
18	5, 17	bitrdi 287	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3060   ⊆ wss 3926   “ cima 5657   Fn wfn 6525  ‘cfv 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-fv 6538

This theorem is referenced by:  imaeqsalvOLD  7356