Theorem f1preimaex 35826

Description: If the image under a one-to-one function exists, then the corresponding preimage also exists. (Contributed by BTernaryTau, 21-Jun-2026.)

Assertion

Ref	Expression
f1preimaex	⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → 𝐶 ∈ V)

Proof of Theorem f1preimaex

Step	Hyp	Ref	Expression
1		df-f1 6571	. . . . 5 ⊢ (𝐹:𝐴⟶1-1𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
2	1	simprbi 503	. . . 4 ⊢ (𝐹:𝐴⟶1-1𝐵 → Fun ◡𝐹)
3		funimaexg 6656	. . . 4 ⊢ ((Fun ◡𝐹 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → (◡𝐹 “ (𝐹 “ 𝐶)) ∈ V)
4	2, 3	sylan 592	. . 3 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → (◡𝐹 “ (𝐹 “ 𝐶)) ∈ V)
5	4	3adant2 1149	. 2 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → (◡𝐹 “ (𝐹 “ 𝐶)) ∈ V)
6		f1imacnv 6871	. . . 4 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶)
7	6	eleq1d 2849	. . 3 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴) → ((◡𝐹 “ (𝐹 “ 𝐶)) ∈ V ↔ 𝐶 ∈ V))
8	7	3adant3 1150	. 2 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → ((◡𝐹 “ (𝐹 “ 𝐶)) ∈ V ↔ 𝐶 ∈ V))
9	5, 8	mpbid 235	1 ⊢ ((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → 𝐶 ∈ V)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ^◡ccnv 5657   “ cima 5661  Fun wfun 6560  ⟶wf 6562  ⟶1-1wf1 6563

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-pr 5398

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-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573

This theorem is used by:  vonf1onprcf1ac  36085