Theorem preimafvelsetpreimafv 48025

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 23	. . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ 𝐴)
2		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
3	2	sneqd 4606	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {(𝐹‘𝑥)} = {(𝐹‘𝑋)})
4	3	imaeq2d 6063	. . . . . 6 ⊢ (𝑥 = 𝑋 → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ {(𝐹‘𝑋)}))
5	4	eqeq2d 2780	. . . . 5 ⊢ (𝑥 = 𝑋 → ((◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)})))
6	5	adantl 486	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) → ((◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)})))
7		eqidd 2770	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)}))
8	1, 6, 7	rspcedvd 3592	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
9	8	3ad2ant3 1151	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
10		fnex 7216	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
11		cnvexg 7920	. . . . 5 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
12		imaexg 7909	. . . . 5 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ V)
13	10, 11, 12	3syl 19	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ V)
14	13	3adant3 1148	. . 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 48021	. . 3 ⊢ ((◡𝐹 “ {(𝐹‘𝑋)}) ∈ V → ((◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)})))
17	14, 16	syl 18	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)})))
18	9, 17	mpbird 260	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095  Vcvv 3463  {csn 4594  ^◡ccnv 5661   “ cima 5665   Fn wfn 6532  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by:  imasetpreimafvbijlemfo  48042  fundcmpsurbijinjpreimafv  48044