Theorem f1preimaex 35713

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–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → 𝐶 ∈ V)

Proof of Theorem f1preimaex

Step	Hyp	Ref	Expression
1		df-f1 6543	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
2	1	simprbi 503	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹)
3		funimaexg 6626	. . . 4 ⊢ ((Fun ◡𝐹 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → (◡𝐹 “ (𝐹 “ 𝐶)) ∈ V)
4	2, 3	sylan 592	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → (◡𝐹 “ (𝐹 “ 𝐶)) ∈ V)
5	4	3adant2 1149	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → (◡𝐹 “ (𝐹 “ 𝐶)) ∈ V)
6		f1imacnv 6841	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶)
7	6	eleq1d 2846	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ((◡𝐹 “ (𝐹 “ 𝐶)) ∈ V ↔ 𝐶 ∈ V))
8	7	3adant3 1150	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → ((◡𝐹 “ (𝐹 “ 𝐶)) ∈ V ↔ 𝐶 ∈ V))
9	5, 8	mpbid 235	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → 𝐶 ∈ V)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899  ^◡ccnv 5650   “ cima 5654  Fun wfun 6532  ⟶wf 6534  –1-1→wf1 6535

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-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  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-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  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-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-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545

This theorem is used by:  vonf1onprcf1ac  35894