Theorem elmapintab 36921

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 6113	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
3	2	elintab 4422	. . 3 ⊢ ((𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥))
4	3	anbi2i 726	. 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 943	. . . . 5 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) ∈ 𝑥 ↔ 𝐴 ∈ 𝐸))
7	6	imbi2d 329	. . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑥) ↔ (𝜑 → 𝐴 ∈ 𝐸)))
8	7	albidv 1836	. . 3 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))
9	8	pm5.32i 667	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥)) ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))
10	1, 4, 9	3bitri 285	1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   ∈ wcel 1977  {cab 2596  ∩ cint 4410  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-int 4411  df-iota 5768  df-fv 5812

This theorem is referenced by:  elcnvintab  36927