Theorem inisegn0a 49767

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 6060	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝐴 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴))
2	1	ibi 270	. 2 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴)
3		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
4	3	eliniseg 6090	. . . 4 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝑥 ∈ (◡𝐹 “ {𝐴}) ↔ 𝑥𝐹𝐴))
5		ne0i 4287	. . . 4 ⊢ (𝑥 ∈ (◡𝐹 “ {𝐴}) → (◡𝐹 “ {𝐴}) ≠ ∅)
6	4, 5	biimtrrdi 257	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝑥𝐹𝐴 → (◡𝐹 “ {𝐴}) ≠ ∅))
7	6	rexlimdvw 3168	. 2 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (∃𝑥 ∈ 𝐵 𝑥𝐹𝐴 → (◡𝐹 “ {𝐴}) ≠ ∅))
8	2, 7	mpd 16	1 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (◡𝐹 “ {𝐴}) ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  ∅c0 4279  {csn 4584   class class class wbr 5103  ^◡ccnv 5654   “ cima 5658

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-ext 2732  ax-sep 5251  ax-pr 5398

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-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  imasubc  50080  imaid  50083