Theorem intimafv 33183

Description: The intersection of an image set, as an indexed intersection of function values. (Contributed by Thierry Arnoux, 15-Jun-2024.)

Assertion

Ref	Expression
intimafv	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ∩ (𝐹 “ 𝐴) = ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem intimafv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfimafn 6940	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦})
2	1	inteqd 4912	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ∩ (𝐹 “ 𝐴) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦})
3		fvex 6891	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
4	3	rgenw 3080	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V
5		iinabrex 33042	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V → ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
6	4, 5	ax-mp 5	. . 3 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
7		eqcom 2767	. . . . . 6 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
8	7	rexbii 3109	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
9	8	abbii 2827	. . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
10	9	inteqi 4911	. . 3 ⊢ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦} = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
11	6, 10	eqtr4i 2786	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦}
12	2, 11	eqtr4di 2813	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ∩ (𝐹 “ 𝐴) = ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  ∩ cint 4907  ∩ ciin 4952  dom cdm 5655   “ cima 5658  Fun wfun 6527  ‘cfv 6533

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  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-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  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-uni 4868  df-int 4908  df-iin 4954  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541

This theorem is used by:  zarclsint  34382  zarcmplem  34391