Theorem imaeqsexvOLD 7297

Description: Obsolete version of rexima 7172 as of 14-Aug-2025. Duplicate version of rexima 7172. (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 3057	. . 3 ⊢ (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐹 “ 𝐵) ∧ 𝜑))
2		fvelimab 6894	. . . . 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 1922	. . 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 3259	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑥∃𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑥 ∧ 𝜑))
7		eqcom 2738	. . . . . . 7 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
8	7	anbi1i 624	. . . . . 6 ⊢ (((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ (𝑥 = (𝐹‘𝑦) ∧ 𝜑))
9	8	exbii 1849	. . . . 5 ⊢ (∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑥(𝑥 = (𝐹‘𝑦) ∧ 𝜑))
10		fvex 6835	. . . . . 6 ⊢ (𝐹‘𝑦) ∈ V
11		imaeqsexvOLD.1	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
12	10, 11	ceqsexv 3486	. . . . 5 ⊢ (∃𝑥(𝑥 = (𝐹‘𝑦) ∧ 𝜑) ↔ 𝜓)
13	9, 12	bitri 275	. . . 4 ⊢ (∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ 𝜓)
14	13	rexbii 3079	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓)
15		r19.41v 3162	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑))
16	15	exbii 1849	. . 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 1541  ∃wex 1780   ∈ wcel 2111  ∃wrex 3056   ⊆ wss 3897   “ cima 5617   Fn wfn 6476  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489

This theorem is referenced by:  imaeqsalvOLD  7298