Theorem elintima 44412

Description: Element of intersection of images. (Contributed by RP, 13-Apr-2020.)

Assertion

Ref	Expression
elintima	⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝑎   𝐵,𝑏   𝑎,𝑏,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦, 𝑎, 𝑏)   𝐵(𝑦, 𝑎)

Proof of Theorem elintima

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3461	. . 3 ⊢ 𝑦 ∈ V
2	1	elint2 4921	. 2 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧)
3		elequ2 2161	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
4	3	ralab2 3662	. . 3 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧 ↔ ∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥))
5		df-rex 3092	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) ↔ ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)))
6	5	imbi1i 352	. . . . . 6 ⊢ ((∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ (∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥))
7		19.23v 1975	. . . . . 6 ⊢ (∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ (∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥))
8		simpr 490	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑥 = (𝑎 “ 𝐵))
9	8	eleq2d 2851	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝑎 “ 𝐵)))
10	9	pm5.74i 274	. . . . . . . 8 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ (𝑎 “ 𝐵)))
11	1	elima 6069	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑎 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏𝑎𝑦)
12		df-br 5112	. . . . . . . . . . 11 ⊢ (𝑏𝑎𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝑎)
13	12	rexbii 3114	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ 𝐵 𝑏𝑎𝑦 ↔ ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
14	11, 13	bitri 278	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑎 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
15	14	imbi2i 339	. . . . . . . 8 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ (𝑎 “ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
16	10, 15	bitri 278	. . . . . . 7 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
17	16	albii 1852	. . . . . 6 ⊢ (∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ ∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
18	6, 7, 17	3bitr2i 302	. . . . 5 ⊢ ((∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
19	18	albii 1852	. . . 4 ⊢ (∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑥∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
20		19.23v 1975	. . . . . . 7 ⊢ (∀𝑥((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ (∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
21		vex 3461	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
22	21	imaex 7913	. . . . . . . . . 10 ⊢ (𝑎 “ 𝐵) ∈ V
23	22	isseti 3475	. . . . . . . . 9 ⊢ ∃𝑥 𝑥 = (𝑎 “ 𝐵)
24		19.42v 1986	. . . . . . . . 9 ⊢ (∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) ↔ (𝑎 ∈ 𝐴 ∧ ∃𝑥 𝑥 = (𝑎 “ 𝐵)))
25	23, 24	mpbiran2 723	. . . . . . . 8 ⊢ (∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) ↔ 𝑎 ∈ 𝐴)
26	25	imbi1i 352	. . . . . . 7 ⊢ ((∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
27	20, 26	bitri 278	. . . . . 6 ⊢ (∀𝑥((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
28	27	albii 1852	. . . . 5 ⊢ (∀𝑎∀𝑥((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎(𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
29		alcom 2197	. . . . 5 ⊢ (∀𝑥∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎∀𝑥((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
30		df-ral 3082	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎(𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
31	28, 29, 30	3bitr4i 306	. . . 4 ⊢ (∀𝑥∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
32	19, 31	bitri 278	. . 3 ⊢ (∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
33	4, 32	bitri 278	. 2 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧 ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
34	2, 33	bitri 278	1 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091  ⟨cop 4597  ∩ cint 4914   class class class wbr 5111   “ cima 5666

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-un 7738

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-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by:  intimass  44413  intimag  44415