Theorem imaelsetpreimafv 43630

Description: The image of an element of the preimage of a function value is the singleton consisting of the function value at one of its elements. (Contributed by AV, 5-Mar-2024.)

Hypothesis

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

Assertion

Ref	Expression
imaelsetpreimafv	⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑋)})

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

Allowed substitution hints:   𝑃(𝑧)   𝑋(𝑧)

Proof of Theorem imaelsetpreimafv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		setpreimafvex.p	. . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2	1	fvelsetpreimafv 43622	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
3		fveq2 6663	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
4	3	sneqd 4572	. . . . . . 7 ⊢ (𝑦 = 𝑥 → {(𝐹‘𝑦)} = {(𝐹‘𝑥)})
5	4	imaeq2d 5922	. . . . . 6 ⊢ (𝑦 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑦)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
6	5	eqeq2d 2831	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
7	6	cbvrexvw 3447	. . . 4 ⊢ (∃𝑦 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
8	2, 7	sylibr 236	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑦 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}))
9	8	3adant3 1127	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → ∃𝑦 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}))
10		imaeq2 5918	. . . . 5 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) → (𝐹 “ 𝑆) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})))
11	10	3ad2ant3 1130	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)})) → (𝐹 “ 𝑆) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})))
12		fnfun 6446	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
13		funimacnv 6428	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})) = ({(𝐹‘𝑦)} ∩ ran 𝐹))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})) = ({(𝐹‘𝑦)} ∩ ran 𝐹))
15	14	3ad2ant1 1128	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})) = ({(𝐹‘𝑦)} ∩ ran 𝐹))
16	15	3ad2ant1 1128	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)})) → (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})) = ({(𝐹‘𝑦)} ∩ ran 𝐹))
17	1	elsetpreimafvbi 43626	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋))))
18		fnfvelrn 6841	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ran 𝐹)
19	18	snssd 4735	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → {(𝐹‘𝑦)} ⊆ ran 𝐹)
20		df-ss 3945	. . . . . . . . . . . 12 ⊢ ({(𝐹‘𝑦)} ⊆ ran 𝐹 ↔ ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑦)})
21	19, 20	sylib 220	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑦)})
22	21	3adant3 1127	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑦)})
23		simp3 1133	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → (𝐹‘𝑦) = (𝐹‘𝑋))
24	23	sneqd 4572	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → {(𝐹‘𝑦)} = {(𝐹‘𝑋)})
25	22, 24	eqtrd 2855	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)})
26	25	3expib 1117	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → ((𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)}))
27	26	3ad2ant1 1128	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → ((𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)}))
28	17, 27	sylbid 242	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑦 ∈ 𝑆 → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)}))
29	28	imp 409	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)})
30	29	3adant3 1127	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)})) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)})
31	11, 16, 30	3eqtrd 2859	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)})) → (𝐹 “ 𝑆) = {(𝐹‘𝑋)})
32	31	rexlimdv3a 3285	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (∃𝑦 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) → (𝐹 “ 𝑆) = {(𝐹‘𝑋)}))
33	9, 32	mpd 15	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑋)})

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113  {cab 2798  ∃wrex 3138   ∩ cin 3928   ⊆ wss 3929  {csn 4560  ^◡ccnv 5547  ran crn 5549   “ cima 5551  Fun wfun 6342   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-sep 5196  ax-nul 5203  ax-pr 5323

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-rab 3146  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  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-fv 6356

This theorem is referenced by:  uniimaelsetpreimafv  43631