Theorem inisegn0a 49023

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 6021	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝐴 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴))
2	1	ibi 267	. 2 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴)
3		vex 3442	. . . . 5 ⊢ 𝑥 ∈ V
4	3	eliniseg 6051	. . . 4 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝑥 ∈ (◡𝐹 “ {𝐴}) ↔ 𝑥𝐹𝐴))
5		ne0i 4291	. . . 4 ⊢ (𝑥 ∈ (◡𝐹 “ {𝐴}) → (◡𝐹 “ {𝐴}) ≠ ∅)
6	4, 5	biimtrrdi 254	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝑥𝐹𝐴 → (◡𝐹 “ {𝐴}) ≠ ∅))
7	6	rexlimdvw 3140	. 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 2113   ≠ wne 2930  ∃wrex 3058  ∅c0 4283  {csn 4578   class class class wbr 5096  ^◡ccnv 5621   “ cima 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635

This theorem is referenced by:  imasubc  49338  imaid  49341