Theorem elsetpreimafveq 47967

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 2773	. . . . 5 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) → ((𝐹‘𝑥) = (𝐹‘𝑋) ↔ (𝐹‘𝑥) = (𝐹‘𝑌)))
2	1	rabbidv 3420	. . . 4 ⊢ ((𝐹‘𝑋) = (𝐹‘𝑌) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑌)})
3	2	adantl 485	. . 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 486	. . . . . 6 ⊢ ((𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) → 𝑆 ∈ 𝑃)
6		simpl 486	. . . . . 6 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅) → 𝑋 ∈ 𝑆)
7	4, 5, 6	3anim123i 1163	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) → (𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆))
8	7	adantr 484	. . . 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 47964	. . . 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 488	. . . . . 6 ⊢ ((𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) → 𝑅 ∈ 𝑃)
13		simpr 488	. . . . . 6 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅) → 𝑌 ∈ 𝑅)
14	4, 12, 13	3anim123i 1163	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) → (𝐹 Fn 𝐴 ∧ 𝑅 ∈ 𝑃 ∧ 𝑌 ∈ 𝑅))
15	14	adantr 484	. . . 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 47964	. . . 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 2806	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) ∧ (𝐹‘𝑋) = (𝐹‘𝑌)) → 𝑆 = 𝑅)
19	18	ex 416	1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑆 = 𝑅))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {cab 2739  ∃wrex 3085  {crab 3413  {csn 4581  ^◡ccnv 5644   “ cima 5648   Fn wfn 6512  ‘cfv 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-fv 6525

This theorem is referenced by:  imasetpreimafvbijlemf1  47974