Theorem elmapintab 44136

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 6876	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
3	2	elintab 4916	. . 3 ⊢ ((𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥))
4	3	anbi2i 632	. 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 544	. . . . 5 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) ∈ 𝑥 ↔ 𝐴 ∈ 𝐸))
7	6	imbi2d 342	. . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑥) ↔ (𝜑 → 𝐴 ∈ 𝐸)))
8	7	albidv 1939	. . 3 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))
9	8	pm5.32i 582	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥)) ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))
10	1, 4, 9	3bitri 299	1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   ∈ wcel 2141  {cab 2739  ∩ cint 4904  ‘cfv 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-sn 4582  df-pr 4584  df-uni 4865  df-int 4905  df-iota 6473  df-fv 6525

This theorem is referenced by:  elcnvintab  44142