Theorem elintima 41261

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 3436	. . 3 ⊢ 𝑦 ∈ V
2	1	elint2 4886	. 2 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧)
3		elequ2 2121	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
4	3	ralab2 3634	. . 3 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧 ↔ ∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥))
5		df-rex 3070	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) ↔ ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)))
6	5	imbi1i 350	. . . . . 6 ⊢ ((∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ (∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥))
7		19.23v 1945	. . . . . 6 ⊢ (∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ (∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥))
8		simpr 485	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑥 = (𝑎 “ 𝐵))
9	8	eleq2d 2824	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝑎 “ 𝐵)))
10	9	pm5.74i 270	. . . . . . . 8 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ (𝑎 “ 𝐵)))
11	1	elima 5974	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑎 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏𝑎𝑦)
12		df-br 5075	. . . . . . . . . . 11 ⊢ (𝑏𝑎𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝑎)
13	12	rexbii 3181	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ 𝐵 𝑏𝑎𝑦 ↔ ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
14	11, 13	bitri 274	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑎 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
15	14	imbi2i 336	. . . . . . . 8 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ (𝑎 “ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
16	10, 15	bitri 274	. . . . . . 7 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
17	16	albii 1822	. . . . . 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 299	. . . . 5 ⊢ ((∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
19	18	albii 1822	. . . 4 ⊢ (∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑥∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
20		19.23v 1945	. . . . . . 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 3436	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
22	21	imaex 7763	. . . . . . . . . 10 ⊢ (𝑎 “ 𝐵) ∈ V
23	22	isseti 3447	. . . . . . . . 9 ⊢ ∃𝑥 𝑥 = (𝑎 “ 𝐵)
24		19.42v 1957	. . . . . . . . 9 ⊢ (∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) ↔ (𝑎 ∈ 𝐴 ∧ ∃𝑥 𝑥 = (𝑎 “ 𝐵)))
25	23, 24	mpbiran2 707	. . . . . . . 8 ⊢ (∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) ↔ 𝑎 ∈ 𝐴)
26	25	imbi1i 350	. . . . . . 7 ⊢ ((∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
27	20, 26	bitri 274	. . . . . 6 ⊢ (∀𝑥((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
28	27	albii 1822	. . . . 5 ⊢ (∀𝑎∀𝑥((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎(𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
29		alcom 2156	. . . . 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 3069	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎(𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
31	28, 29, 30	3bitr4i 303	. . . 4 ⊢ (∀𝑥∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
32	19, 31	bitri 274	. . 3 ⊢ (∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
33	4, 32	bitri 274	. 2 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧 ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
34	2, 33	bitri 274	1 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065  ⟨cop 4567  ∩ cint 4879   class class class wbr 5074   “ cima 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602

This theorem is referenced by:  intimass  41262  intimag  41264