Theorem elsetpreimafvbi 48472

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 7059	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥))))
2		fniniseg 7059	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥))))
3		eqeq2 2773	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑋) → ((𝐹‘𝑌) = (𝐹‘𝑥) ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
4	3	anbi2d 642	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑋) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
5	4	eqcoms 2769	. . . . . . . . 9 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑥) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
6	2, 5	sylan9bb 519	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
7	6	ex 418	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ((𝐹‘𝑋) = (𝐹‘𝑥) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
8	7	adantld 496	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
9	1, 8	sylbid 243	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
10		eleq2 2850	. . . . . 6 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
11		eleq2 2850	. . . . . . 7 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ 𝑆 ↔ 𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
12	11	bibi1d 346	. . . . . 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 347	. . . . 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 249	. . . 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 48466	. . 3 ⊢ (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
18	15, 17	syl11 34	. 2 ⊢ (𝐹 Fn 𝐴 → (𝑆 ∈ 𝑃 → (𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))))
19	18	3imp 1128	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2739  ∃wrex 3087  {csn 4584  ^◡ccnv 5650   “ cima 5654   Fn wfn 6533  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by:  elsetpreimafveqfv  48473  eqfvelsetpreimafv  48474  elsetpreimafvrab  48475  imaelsetpreimafv  48476