Theorem imaelsetpreimafv 48002

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 47994	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
3		fveq2 6868	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
4	3	sneqd 4595	. . . . . . 7 ⊢ (𝑦 = 𝑥 → {(𝐹‘𝑦)} = {(𝐹‘𝑥)})
5	4	imaeq2d 6050	. . . . . 6 ⊢ (𝑦 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑦)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
6	5	eqeq2d 2774	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
7	6	cbvrexvw 3242	. . . 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 1146	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → ∃𝑦 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}))
10		imaeq2 6046	. . . . 5 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) → (𝐹 “ 𝑆) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})))
11	10	3ad2ant3 1149	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)})) → (𝐹 “ 𝑆) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})))
12		fnfun 6622	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
13		funimacnv 6603	. . . . . . 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 1147	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})) = ({(𝐹‘𝑦)} ∩ ran 𝐹))
16	15	3ad2ant1 1147	. . . 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 47998	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋))))
18		fnfvelrn 7062	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ran 𝐹)
19	18	snssd 4746	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → {(𝐹‘𝑦)} ⊆ ran 𝐹)
20		dfss2 3923	. . . . . . . . . . . 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 1146	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑦)})
23		simp3 1152	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → (𝐹‘𝑦) = (𝐹‘𝑋))
24	23	sneqd 4595	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → {(𝐹‘𝑦)} = {(𝐹‘𝑋)})
25	22, 24	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)})
26	25	3expib 1136	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → ((𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)}))
27	26	3ad2ant1 1147	. . . . . . 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 410	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)})
30	29	3adant3 1146	. . . 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 2802	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)})) → (𝐹 “ 𝑆) = {(𝐹‘𝑋)})
32	31	rexlimdv3a 3168	. 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 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  {cab 2741  ∃wrex 3087   ∩ cin 3904   ⊆ wss 3905  {csn 4583  ^◡ccnv 5647  ran crn 5649   “ cima 5651  Fun wfun 6516   Fn wfn 6517  ‘cfv 6522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-fv 6530

This theorem is referenced by:  uniimaelsetpreimafv  48003