Theorem imaeqsexv 33690

Description: Substitute a function value into an existential quantifier over an image. (Contributed by Scott Fenton, 27-Sep-2024.)

Hypothesis

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

Assertion

Ref	Expression
imaeqsexv	⊢ ((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 imaeqsexv

Step	Hyp	Ref	Expression
1		df-rex 3070	. . 3 ⊢ (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐹 “ 𝐵) ∧ 𝜑))
2		fvelimab 6841	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
3	2	anbi1d 630	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑥 ∈ (𝐹 “ 𝐵) ∧ 𝜑) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑)))
4	3	exbidv 1924	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥(𝑥 ∈ (𝐹 “ 𝐵) ∧ 𝜑) ↔ ∃𝑥(∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑)))
5	1, 4	syl5bb 283	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑥(∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ∧ 𝜑)))
6		rexcom4 3233	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑥∃𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑥 ∧ 𝜑))
7		eqcom 2745	. . . . . . 7 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
8	7	anbi1i 624	. . . . . 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 6787	. . . . . 6 ⊢ (𝐹‘𝑦) ∈ V
11		imaeqsex.1	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
12	10, 11	ceqsexv 3479	. . . . 5 ⊢ (∃𝑥(𝑥 = (𝐹‘𝑦) ∧ 𝜑) ↔ 𝜓)
13	9, 12	bitri 274	. . . 4 ⊢ (∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ 𝜓)
14	13	rexbii 3181	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥((𝐹‘𝑦) = 𝑥 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 𝜓)
15		r19.41v 3276	. . . 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 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065   ⊆ wss 3887   “ cima 5592   Fn wfn 6428  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  imaeqsalv  33691