Theorem elintima 43225

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 3465	. . 3 ⊢ 𝑦 ∈ V
2	1	elint2 4957	. 2 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧)
3		elequ2 2113	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
4	3	ralab2 3689	. . 3 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧 ↔ ∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥))
5		df-rex 3060	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) ↔ ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)))
6	5	imbi1i 348	. . . . . 6 ⊢ ((∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ (∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥))
7		19.23v 1937	. . . . . 6 ⊢ (∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ (∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥))
8		simpr 483	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑥 = (𝑎 “ 𝐵))
9	8	eleq2d 2811	. . . . . . . . 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 6069	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑎 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏𝑎𝑦)
12		df-br 5150	. . . . . . . . . . 11 ⊢ (𝑏𝑎𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝑎)
13	12	rexbii 3083	. . . . . . . . . 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 335	. . . . . . . 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 1813	. . . . . 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 298	. . . . 5 ⊢ ((∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
19	18	albii 1813	. . . 4 ⊢ (∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑥∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
20		19.23v 1937	. . . . . . 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 3465	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
22	21	imaex 7922	. . . . . . . . . 10 ⊢ (𝑎 “ 𝐵) ∈ V
23	22	isseti 3477	. . . . . . . . 9 ⊢ ∃𝑥 𝑥 = (𝑎 “ 𝐵)
24		19.42v 1949	. . . . . . . . 9 ⊢ (∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) ↔ (𝑎 ∈ 𝐴 ∧ ∃𝑥 𝑥 = (𝑎 “ 𝐵)))
25	23, 24	mpbiran2 708	. . . . . . . 8 ⊢ (∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) ↔ 𝑎 ∈ 𝐴)
26	25	imbi1i 348	. . . . . . 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 1813	. . . . 5 ⊢ (∀𝑎∀𝑥((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎(𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
29		alcom 2148	. . . . 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 3051	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎(𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
31	28, 29, 30	3bitr4i 302	. . . 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 394  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∀wral 3050  ∃wrex 3059  ⟨cop 4636  ∩ cint 4950   class class class wbr 5149   “ cima 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691

This theorem is referenced by:  intimass  43226  intimag  43228