Theorem imaelsetpreimafv 47855

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 47847	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
3		fveq2 6840	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
4	3	sneqd 4579	. . . . . . 7 ⊢ (𝑦 = 𝑥 → {(𝐹‘𝑦)} = {(𝐹‘𝑥)})
5	4	imaeq2d 6025	. . . . . 6 ⊢ (𝑦 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑦)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
6	5	eqeq2d 2747	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
7	6	cbvrexvw 3216	. . . 4 ⊢ (∃𝑦 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
8	2, 7	sylibr 234	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑦 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}))
9	8	3adant3 1133	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → ∃𝑦 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}))
10		imaeq2 6021	. . . . 5 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) → (𝐹 “ 𝑆) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})))
11	10	3ad2ant3 1136	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)})) → (𝐹 “ 𝑆) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})))
12		fnfun 6598	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
13		funimacnv 6579	. . . . . . 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 1134	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝐹 “ (◡𝐹 “ {(𝐹‘𝑦)})) = ({(𝐹‘𝑦)} ∩ ran 𝐹))
16	15	3ad2ant1 1134	. . . 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 47851	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋))))
18		fnfvelrn 7032	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ran 𝐹)
19	18	snssd 4730	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → {(𝐹‘𝑦)} ⊆ ran 𝐹)
20		dfss2 3907	. . . . . . . . . . . 12 ⊢ ({(𝐹‘𝑦)} ⊆ ran 𝐹 ↔ ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑦)})
21	19, 20	sylib 218	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑦)})
22	21	3adant3 1133	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑦)})
23		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → (𝐹‘𝑦) = (𝐹‘𝑋))
24	23	sneqd 4579	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → {(𝐹‘𝑦)} = {(𝐹‘𝑋)})
25	22, 24	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)})
26	25	3expib 1123	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → ((𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)}))
27	26	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → ((𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)}))
28	17, 27	sylbid 240	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑦 ∈ 𝑆 → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)}))
29	28	imp 406	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆) → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)})
30	29	3adant3 1133	. . . 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 2775	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)})) → (𝐹 “ 𝑆) = {(𝐹‘𝑋)})
32	31	rexlimdv3a 3142	. 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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2714  ∃wrex 3061   ∩ cin 3888   ⊆ wss 3889  {csn 4567  ^◡ccnv 5630  ran crn 5632   “ cima 5634  Fun wfun 6492   Fn wfn 6493  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506

This theorem is referenced by:  uniimaelsetpreimafv  47856