Theorem elsetpreimafvbi 46049

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 7061	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥))))
2		fniniseg 7061	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥))))
3		eqeq2 2744	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑋) → ((𝐹‘𝑌) = (𝐹‘𝑥) ↔ (𝐹‘𝑌) = (𝐹‘𝑋)))
4	3	anbi2d 629	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑋) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
5	4	eqcoms 2740	. . . . . . . . 9 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑥) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑥)) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
6	2, 5	sylan9bb 510	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
7	6	ex 413	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → ((𝐹‘𝑋) = (𝐹‘𝑥) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
8	7	adantld 491	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = (𝐹‘𝑥)) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
9	1, 8	sylbid 239	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑌 ∈ (◡𝐹 “ {(𝐹‘𝑥)}) ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)))))
10		eleq2 2822	. . . . . 6 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑥)})))
11		eleq2 2822	. . . . . . 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 245	. . . 4 ⊢ (𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}) → (𝐹 Fn 𝐴 → (𝑋 ∈ 𝑆 → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))))
15	14	rexlimivw 3151	. . 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 46043	. . 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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  ∃wrex 3070  {csn 4628  ^◡ccnv 5675   “ cima 5679   Fn wfn 6538  ‘cfv 6543

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by:  elsetpreimafveqfv  46050  eqfvelsetpreimafv  46051  elsetpreimafvrab  46052  imaelsetpreimafv  46053