Theorem elsetpreimafveq 44737

Description: If two preimages of function values contain elements with identical function values, then both preimages are equal. (Contributed by AV, 8-Mar-2024.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem elsetpreimafveq

Step	Hyp	Ref	Expression
1		eqeq2 2750	. . . . 5 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) → ((𝐹‘𝑥) = (𝐹‘𝑋) ↔ (𝐹‘𝑥) = (𝐹‘𝑌)))
2	1	rabbidv 3404	. . . 4 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑌)})
3	2	adantl 481	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑌)})
4		id 22	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn 𝐴)
5		simpl 482	. . . . . 6 ⊢ ((𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) → 𝑆 ∈ 𝑃)
6		simpl 482	. . . . . 6 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅) → 𝑋 ∈ 𝑆)
7	4, 5, 6	3anim123i 1149	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) → (𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆))
8	7	adantr 480	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → (𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆))
9		setpreimafvex.p	. . . . 5 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
10	9	elsetpreimafvrab 44734	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → 𝑆 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)})
11	8, 10	syl 17	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → 𝑆 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)})
12		simpr 484	. . . . . 6 ⊢ ((𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) → 𝑅 ∈ 𝑃)
13		simpr 484	. . . . . 6 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅) → 𝑌 ∈ 𝑅)
14	4, 12, 13	3anim123i 1149	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) → (𝐹 Fn 𝐴 ∧ 𝑅 ∈ 𝑃 ∧ 𝑌 ∈ 𝑅))
15	14	adantr 480	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → (𝐹 Fn 𝐴 ∧ 𝑅 ∈ 𝑃 ∧ 𝑌 ∈ 𝑅))
16	9	elsetpreimafvrab 44734	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑅 ∈ 𝑃 ∧ 𝑌 ∈ 𝑅) → 𝑅 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑌)})
17	15, 16	syl 17	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → 𝑅 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑌)})
18	3, 11, 17	3eqtr4d 2788	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → 𝑆 = 𝑅)
19	18	ex 412	1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑆 = 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  {crab 3067  {csn 4558  ^◡ccnv 5579   “ cima 5583   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by:  imasetpreimafvbijlemf1  44744