Theorem elintima 38469

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 3354	. . 3 ⊢ 𝑦 ∈ V
2	1	elint2 4619	. 2 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧)
3		elequ2 2159	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
4	3	ralab2 3523	. . 3 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧 ↔ ∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥))
5		df-rex 3067	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) ↔ ∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)))
6	5	imbi1i 338	. . . . . 6 ⊢ ((∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ (∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥))
7		19.23v 2023	. . . . . 6 ⊢ (∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ (∃𝑎(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥))
8		simpr 471	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑥 = (𝑎 “ 𝐵))
9	8	eleq2d 2836	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ (𝑎 “ 𝐵)))
10	9	pm5.74i 260	. . . . . . . 8 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ (𝑎 “ 𝐵)))
11	1	elima 5611	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑎 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏𝑎𝑦)
12		df-br 4788	. . . . . . . . . . 11 ⊢ (𝑏𝑎𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝑎)
13	12	rexbii 3189	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ 𝐵 𝑏𝑎𝑦 ↔ ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
14	11, 13	bitri 264	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑎 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
15	14	imbi2i 325	. . . . . . . 8 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ (𝑎 “ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
16	10, 15	bitri 264	. . . . . . 7 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → 𝑦 ∈ 𝑥) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
17	16	albii 1895	. . . . . 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 288	. . . . 5 ⊢ ((∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
19	18	albii 1895	. . . 4 ⊢ (∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑥∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
20		19.23v 2023	. . . . . . 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 3354	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
22		imaexg 7254	. . . . . . . . . . 11 ⊢ (𝑎 ∈ V → (𝑎 “ 𝐵) ∈ V)
23	21, 22	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑎 “ 𝐵) ∈ V
24	23	isseti 3361	. . . . . . . . 9 ⊢ ∃𝑥 𝑥 = (𝑎 “ 𝐵)
25		19.42v 2033	. . . . . . . . 9 ⊢ (∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) ↔ (𝑎 ∈ 𝐴 ∧ ∃𝑥 𝑥 = (𝑎 “ 𝐵)))
26	24, 25	mpbiran2 689	. . . . . . . 8 ⊢ (∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) ↔ 𝑎 ∈ 𝐴)
27	26	imbi1i 338	. . . . . . 7 ⊢ ((∃𝑥(𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
28	20, 27	bitri 264	. . . . . 6 ⊢ (∀𝑥((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ (𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
29	28	albii 1895	. . . . 5 ⊢ (∀𝑎∀𝑥((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎(𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
30		alcom 2193	. . . . 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))
31		df-ral 3066	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 ↔ ∀𝑎(𝑎 ∈ 𝐴 → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎))
32	29, 30, 31	3bitr4i 292	. . . 4 ⊢ (∀𝑥∀𝑎((𝑎 ∈ 𝐴 ∧ 𝑥 = (𝑎 “ 𝐵)) → ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎) ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
33	19, 32	bitri 264	. . 3 ⊢ (∀𝑥(∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵) → 𝑦 ∈ 𝑥) ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
34	4, 33	bitri 264	. 2 ⊢ (∀𝑧 ∈ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}𝑦 ∈ 𝑧 ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
35	2, 34	bitri 264	1 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1629   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757  ∀wral 3061  ∃wrex 3062  Vcvv 3351  ⟨cop 4323  ∩ cint 4612   class class class wbr 4787   “ cima 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263

This theorem is referenced by:  intimass  38470  intimag  38472