Theorem inisegn0a 49499

Description: The inverse image of a singleton subset of an image is non-empty. (Contributed by Zhi Wang, 7-Nov-2025.)

Assertion

Ref	Expression
inisegn0a	⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (◡𝐹 “ {𝐴}) ≠ ∅)

Proof of Theorem inisegn0a

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elimag 6067	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝐴 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴))
2	1	ibi 270	. 2 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴)
3		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
4	3	eliniseg 6097	. . . 4 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝑥 ∈ (◡𝐹 “ {𝐴}) ↔ 𝑥𝐹𝐴))
5		ne0i 4302	. . . 4 ⊢ (𝑥 ∈ (◡𝐹 “ {𝐴}) → (◡𝐹 “ {𝐴}) ≠ ∅)
6	4, 5	biimtrrdi 257	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝑥𝐹𝐴 → (◡𝐹 “ {𝐴}) ≠ ∅))
7	6	rexlimdvw 3177	. 2 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (∃𝑥 ∈ 𝐵 𝑥𝐹𝐴 → (◡𝐹 “ {𝐴}) ≠ ∅))
8	2, 7	mpd 16	1 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (◡𝐹 “ {𝐴}) ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  ∅c0 4294  {csn 4594   class class class wbr 5113  ^◡ccnv 5661   “ cima 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675

This theorem is referenced by:  imasubc  49814  imaid  49817