Theorem preimafvelsetpreimafv 46897

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 6900	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
3	2	sneqd 4644	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {(𝐹‘𝑥)} = {(𝐹‘𝑋)})
4	3	imaeq2d 6068	. . . . . 6 ⊢ (𝑥 = 𝑋 → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ {(𝐹‘𝑋)}))
5	4	eqeq2d 2736	. . . . 5 ⊢ (𝑥 = 𝑋 → ((◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)})))
6	5	adantl 480	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) → ((◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)})))
7		eqidd 2726	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)}))
8	1, 6, 7	rspcedvd 3609	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
9	8	3ad2ant3 1132	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
10		fnex 7233	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
11		cnvexg 7936	. . . . 5 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
12		imaexg 7925	. . . . 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 1129	. . 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 46893	. . 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 256	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2702  ∃wrex 3059  Vcvv 3461  {csn 4632  ^◡ccnv 5680   “ cima 5684   Fn wfn 6548  ‘cfv 6553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561

This theorem is referenced by:  imasetpreimafvbijlemfo  46914  fundcmpsurbijinjpreimafv  46916