Theorem imaelsetpreimafv 46063

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 46055	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
3		fveq2 6892	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
4	3	sneqd 4641	. . . . . . 7 ⊢ (𝑦 = 𝑥 → {(𝐹‘𝑦)} = {(𝐹‘𝑥)})
5	4	imaeq2d 6060	. . . . . 6 ⊢ (𝑦 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑦)}) = (◡𝐹 “ {(𝐹‘𝑥)}))
6	5	eqeq2d 2744	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
7	6	cbvrexvw 3236	. . . 4 ⊢ (∃𝑦 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
8	2, 7	sylibr 233	. . 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 6056	. . . . 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 6650	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
13		funimacnv 6630	. . . . . . 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 46059	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋))))
18		fnfvelrn 7083	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ran 𝐹)
19	18	snssd 4813	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → {(𝐹‘𝑦)} ⊆ ran 𝐹)
20		df-ss 3966	. . . . . . . . . . . 12 ⊢ ({(𝐹‘𝑦)} ⊆ ran 𝐹 ↔ ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑦)})
21	19, 20	sylib 217	. . . . . . . . . . 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 4641	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = (𝐹‘𝑋)) → {(𝐹‘𝑦)} = {(𝐹‘𝑋)})
25	22, 24	eqtrd 2773	. . . . . . . . 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 239	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑦 ∈ 𝑆 → ({(𝐹‘𝑦)} ∩ ran 𝐹) = {(𝐹‘𝑋)}))
29	28	imp 408	. . . . 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 2777	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) ∧ 𝑦 ∈ 𝑆 ∧ 𝑆 = (◡𝐹 “ {(𝐹‘𝑦)})) → (𝐹 “ 𝑆) = {(𝐹‘𝑋)})
32	31	rexlimdv3a 3160	. 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071   ∩ cin 3948   ⊆ wss 3949  {csn 4629  ^◡ccnv 5676  ran crn 5678   “ cima 5680  Fun wfun 6538   Fn wfn 6539  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552

This theorem is referenced by:  uniimaelsetpreimafv  46064