Theorem elmapintab 43609

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 6919	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
3	2	elintab 4958	. . 3 ⊢ ((𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥))
4	3	anbi2i 623	. 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 536	. . . . 5 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) ∈ 𝑥 ↔ 𝐴 ∈ 𝐸))
7	6	imbi2d 340	. . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑥) ↔ (𝜑 → 𝐴 ∈ 𝐸)))
8	7	albidv 1920	. . 3 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))
9	8	pm5.32i 574	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥)) ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))
10	1, 4, 9	3bitri 297	1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2108  {cab 2714  ∩ cint 4946  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-int 4947  df-iota 6514  df-fv 6569

This theorem is referenced by:  elcnvintab  43615