Theorem imaeqsexvOLD 7307

Description: Obsolete version of rexima 7182 as of 14-Aug-2025. Duplicate version of rexima 7182. (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 3060	. . 3 ⊢ (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐹 “ 𝐵) ∧ 𝜑))
2		fvelimab 6901	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
3	2	anbi1d 632	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ 𝐵) ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑)))
4	3	exbidv 1923	. . 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 3262	. . 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 625	. . . . . 6 ⊢ (((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ (𝑥 = (𝐹‘𝑦) ∧ 𝜑))
9	8	exbii 1850	. . . . 5 ⊢ (∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝑥 = (𝐹‘𝑦) ∧ 𝜑))
10		fvex 6842	. . . . . 6 ⊢ (𝐹‘𝑦) ∈ V
11		imaeqsexvOLD.1	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
12	10, 11	ceqsexv 3476	. . . . 5 ⊢ (∃𝑥(𝑥 = (𝐹‘𝑦) ∧ 𝜑) ↔ 𝜓)
13	9, 12	bitri 275	. . . 4 ⊢ (∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ 𝜓)
14	13	rexbii 3082	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓)
15		r19.41v 3165	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑))
16	15	exbii 1850	. . 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 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3059   ⊆ wss 3885   “ cima 5623   Fn wfn 6482  ‘cfv 6487

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-fv 6495

This theorem is referenced by:  imaeqsalvOLD  7308