Theorem preimafvelsetpreimafv 43623

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 6663	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
3	2	sneqd 4572	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {(𝐹‘𝑥)} = {(𝐹‘𝑋)})
4	3	imaeq2d 5922	. . . . . 6 ⊢ (𝑥 = 𝑋 → (◡𝐹 “ {(𝐹‘𝑥)}) = (◡𝐹 “ {(𝐹‘𝑋)}))
5	4	eqeq2d 2831	. . . . 5 ⊢ (𝑥 = 𝑋 → ((◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)})))
6	5	adantl 484	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑥 = 𝑋) → ((◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)})))
7		eqidd 2821	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑋)}))
8	1, 6, 7	rspcedvd 3623	. . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
9	8	3ad2ant3 1130	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 (◡𝐹 “ {(𝐹‘𝑋)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
10		fnex 6973	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
11		cnvexg 7622	. . . . 5 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
12		imaexg 7613	. . . . 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 1127	. . 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 43619	. . 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 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113  {cab 2798  ∃wrex 3138  Vcvv 3491  {csn 4560  ^◡ccnv 5547   “ cima 5551   Fn wfn 6343  ‘cfv 6348

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356

This theorem is referenced by:  imasetpreimafvbijlemfo  43640  fundcmpsurbijinjpreimafv  43642