Theorem elmapintab 44184

Description: Two ways to say a set is an element of mapped intersection of a class. Here 𝐹 maps elements of 𝐶 to elements of ∩ {𝑥 ∣ 𝜑} or 𝑥. (Contributed by RP, 19-Aug-2020.)

Hypotheses

Ref	Expression
elmapintab.1	⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑}))
elmapintab.2	⊢ (𝐴 ∈ 𝐸 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ 𝑥))

Assertion

Ref	Expression
elmapintab	⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐸(𝑥)

Proof of Theorem elmapintab

Step	Hyp	Ref	Expression
1		elmapintab.1	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑}))
2		fvex 6884	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
3	2	elintab 4920	. . 3 ⊢ ((𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥))
4	3	anbi2i 634	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥)))
5		elmapintab.2	. . . . . 6 ⊢ (𝐴 ∈ 𝐸 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ 𝑥))
6	5	baibr 545	. . . . 5 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) ∈ 𝑥 ↔ 𝐴 ∈ 𝐸))
7	6	imbi2d 343	. . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑥) ↔ (𝜑 → 𝐴 ∈ 𝐸)))
8	7	albidv 1943	. . 3 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))
9	8	pm5.32i 584	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥)) ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))
10	1, 4, 9	3bitri 300	1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561   ∈ wcel 2145  {cab 2743  ∩ cint 4908  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-sn 4586  df-pr 4588  df-uni 4869  df-int 4909  df-iota 6481  df-fv 6533

This theorem is referenced by:  elcnvintab  44190