Theorem inisegn0a 48817

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 6024	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝐴 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴))
2	1	ibi 267	. 2 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴)
3		vex 3448	. . . . 5 ⊢ 𝑥 ∈ V
4	3	eliniseg 6054	. . . 4 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝑥 ∈ (◡𝐹 “ {𝐴}) ↔ 𝑥𝐹𝐴))
5		ne0i 4300	. . . 4 ⊢ (𝑥 ∈ (◡𝐹 “ {𝐴}) → (◡𝐹 “ {𝐴}) ≠ ∅)
6	4, 5	biimtrrdi 254	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝑥𝐹𝐴 → (◡𝐹 “ {𝐴}) ≠ ∅))
7	6	rexlimdvw 3139	. 2 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (∃𝑥 ∈ 𝐵 𝑥𝐹𝐴 → (◡𝐹 “ {𝐴}) ≠ ∅))
8	2, 7	mpd 15	1 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (◡𝐹 “ {𝐴}) ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  ∅c0 4292  {csn 4585   class class class wbr 5102  ^◡ccnv 5630   “ cima 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  imasubc  49133  imaid  49136