Theorem elsetpreimafvbi 47851

Description: An element of the preimage of a function value is an element of the domain of the function with the same value as another element of the preimage. (Contributed by AV, 9-Mar-2024.)

Hypothesis

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

Assertion

Ref	Expression
elsetpreimafvbi	⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))

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

Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑋(𝑧)   𝑌(𝑧)

Proof of Theorem elsetpreimafvbi

Step	Hyp	Ref	Expression
1		fniniseg 7012	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥))))
2		fniniseg 7012	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥))))
3		eqeq2 2748	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑋) → ((𝐹‘𝑌) = (𝐹‘𝑥) ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
4	3	anbi2d 631	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑋) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
5	4	eqcoms 2744	. . . . . . . . 9 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑥) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
6	2, 5	sylan9bb 509	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
7	6	ex 412	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ((𝐹‘𝑋) = (𝐹‘𝑥) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
8	7	adantld 490	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
9	1, 8	sylbid 240	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
10		eleq2 2825	. . . . . 6 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
11		eleq2 2825	. . . . . . 7 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ 𝑆 ↔ 𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
12	11	bibi1d 343	. . . . . 6 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → ((𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))) ↔ (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
13	10, 12	imbi12d 344	. . . . 5 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → ((𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))) ↔ (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))))
14	9, 13	imbitrrid 246	. . . 4 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐹 Fn 𝐴 → (𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))))
15	14	rexlimivw 3134	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐹 Fn 𝐴 → (𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))))
16		setpreimafvex.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
17	16	elsetpreimafv 47845	. . 3 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
18	15, 17	syl11 33	. 2 ⊢ (𝐹 Fn 𝐴 → (𝑆 ∈ 𝑃 → (𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))))
19	18	3imp 1111	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2714  ∃wrex 3061  {csn 4567  ^◡ccnv 5630   “ cima 5634   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:  elsetpreimafveqfv  47852  eqfvelsetpreimafv  47853  elsetpreimafvrab  47854  imaelsetpreimafv  47855