Theorem preimafvelsetpreimafv 47994

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 6867	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
3	2	sneqd 4594	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {(𝐹‘𝑥)} = {(𝐹‘𝑋)})
4	3	imaeq2d 6049	. . . . . 6 ⊢ (𝑥 = 𝑋 → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ {(𝐹‘𝑋)}))
5	4	eqeq2d 2773	. . . . 5 ⊢ (𝑥 = 𝑋 → ((◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)})))
6	5	adantl 485	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) → ((◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)})))
7		eqidd 2763	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)}))
8	1, 6, 7	rspcedvd 3583	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
9	8	3ad2ant3 1148	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
10		fnex 7201	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
11		cnvexg 7905	. . . . 5 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
12		imaexg 7894	. . . . 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 1145	. . 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 47990	. . 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 259	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  {cab 2740  ∃wrex 3086  Vcvv 3454  {csn 4582  ^◡ccnv 5646   “ cima 5650   Fn wfn 6516  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529

This theorem is referenced by:  imasetpreimafvbijlemfo  48011  fundcmpsurbijinjpreimafv  48013