Theorem inisegn0a 49457

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 6053	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝐴 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴))
2	1	ibi 269	. 2 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴)
3		vex 3458	. . . . 5 ⊢ 𝑥 ∈ V
4	3	eliniseg 6083	. . . 4 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝑥 ∈ (◡𝐹 “ {𝐴}) ↔ 𝑥𝐹𝐴))
5		ne0i 4293	. . . 4 ⊢ (𝑥 ∈ (◡𝐹 “ {𝐴}) → (◡𝐹 “ {𝐴}) ≠ ∅)
6	4, 5	biimtrrdi 256	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝑥𝐹𝐴 → (◡𝐹 “ {𝐴}) ≠ ∅))
7	6	rexlimdvw 3168	. 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 2142   ≠ wne 2957  ∃wrex 3086  ∅c0 4285  {csn 4582   class class class wbr 5100  ^◡ccnv 5646   “ cima 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660

This theorem is referenced by:  imasubc  49772  imaid  49775