Theorem elsetpreimafvbi 47998

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 7042	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥))))
2		fniniseg 7042	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥))))
3		eqeq2 2775	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑋) → ((𝐹‘𝑌) = (𝐹‘𝑥) ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
4	3	anbi2d 639	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑋) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
5	4	eqcoms 2771	. . . . . . . . 9 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑥) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
6	2, 5	sylan9bb 517	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
7	6	ex 416	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ((𝐹‘𝑋) = (𝐹‘𝑥) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
8	7	adantld 494	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
9	1, 8	sylbid 242	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
10		eleq2 2852	. . . . . 6 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
11		eleq2 2852	. . . . . . 7 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ 𝑆 ↔ 𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
12	11	bibi1d 345	. . . . . 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 346	. . . . 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 248	. . . 4 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐹 Fn 𝐴 → (𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))))
15	14	rexlimivw 3160	. . 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 47992	. . 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 1124	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  {cab 2741  ∃wrex 3087  {csn 4583  ^◡ccnv 5647   “ cima 5651   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:  elsetpreimafveqfv  47999  eqfvelsetpreimafv  48000  elsetpreimafvrab  48001  imaelsetpreimafv  48002