Theorem preimafvelsetpreimafv 47393

Description: The preimage of a function value is an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)

Hypothesis

Ref	Expression
setpreimafvex.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}

Assertion

Ref	Expression
preimafvelsetpreimafv	⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃)

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑋,𝑧

Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem preimafvelsetpreimafv

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ 𝐴)
2		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
3	2	sneqd 4604	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {(𝐹‘𝑥)} = {(𝐹‘𝑋)})
4	3	imaeq2d 6034	. . . . . 6 ⊢ (𝑥 = 𝑋 → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ {(𝐹‘𝑋)}))
5	4	eqeq2d 2741	. . . . 5 ⊢ (𝑥 = 𝑋 → ((◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)})))
6	5	adantl 481	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) → ((◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)})))
7		eqidd 2731	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)}))
8	1, 6, 7	rspcedvd 3593	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
9	8	3ad2ant3 1135	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
10		fnex 7194	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
11		cnvexg 7903	. . . . 5 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
12		imaexg 7892	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ V)
13	10, 11, 12	3syl 18	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ V)
14	13	3adant3 1132	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ V)
15		setpreimafvex.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
16	15	elsetpreimafvb 47389	. . 3 ⊢ ((◡𝐹 “ {(𝐹‘𝑋)}) ∈ V → ((◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)})))
17	14, 16	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)})))
18	9, 17	mpbird 257	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2708  ∃wrex 3054  Vcvv 3450  {csn 4592  ^◡ccnv 5640   “ cima 5644   Fn wfn 6509  ‘cfv 6514

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522

This theorem is referenced by:  imasetpreimafvbijlemfo  47410  fundcmpsurbijinjpreimafv  47412