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Theorem elmapintab 40467
 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 (𝐴𝐵 ↔ (𝐴𝐶 ∧ (𝐹𝐴) ∈ {𝑥𝜑}))
elmapintab.2 (𝐴𝐸 ↔ (𝐴𝐶 ∧ (𝐹𝐴) ∈ 𝑥))
Assertion
Ref Expression
elmapintab (𝐴𝐵 ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑𝐴𝐸)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐸(𝑥)

Proof of Theorem elmapintab
StepHypRef Expression
1 elmapintab.1 . 2 (𝐴𝐵 ↔ (𝐴𝐶 ∧ (𝐹𝐴) ∈ {𝑥𝜑}))
2 fvex 6668 . . . 4 (𝐹𝐴) ∈ V
32elintab 4853 . . 3 ((𝐹𝐴) ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥))
43anbi2i 625 . 2 ((𝐴𝐶 ∧ (𝐹𝐴) ∈ {𝑥𝜑}) ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥)))
5 elmapintab.2 . . . . . 6 (𝐴𝐸 ↔ (𝐴𝐶 ∧ (𝐹𝐴) ∈ 𝑥))
65baibr 540 . . . . 5 (𝐴𝐶 → ((𝐹𝐴) ∈ 𝑥𝐴𝐸))
76imbi2d 344 . . . 4 (𝐴𝐶 → ((𝜑 → (𝐹𝐴) ∈ 𝑥) ↔ (𝜑𝐴𝐸)))
87albidv 1921 . . 3 (𝐴𝐶 → (∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥) ↔ ∀𝑥(𝜑𝐴𝐸)))
98pm5.32i 578 . 2 ((𝐴𝐶 ∧ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥)) ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑𝐴𝐸)))
101, 4, 93bitri 300 1 (𝐴𝐵 ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑𝐴𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   ∈ wcel 2111  {cab 2776  ∩ cint 4842  ‘cfv 6332 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5178 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-sn 4529  df-pr 4531  df-uni 4805  df-int 4843  df-iota 6291  df-fv 6340 This theorem is referenced by:  elcnvintab  40473
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