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Theorem elmapintab 38428
 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 6342 . . . 4 (𝐹𝐴) ∈ V
32elintab 4622 . . 3 ((𝐹𝐴) ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥))
43anbi2i 609 . 2 ((𝐴𝐶 ∧ (𝐹𝐴) ∈ {𝑥𝜑}) ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥)))
5 elmapintab.2 . . . . . 6 (𝐴𝐸 ↔ (𝐴𝐶 ∧ (𝐹𝐴) ∈ 𝑥))
65baibr 526 . . . . 5 (𝐴𝐶 → ((𝐹𝐴) ∈ 𝑥𝐴𝐸))
76imbi2d 329 . . . 4 (𝐴𝐶 → ((𝜑 → (𝐹𝐴) ∈ 𝑥) ↔ (𝜑𝐴𝐸)))
87albidv 2001 . . 3 (𝐴𝐶 → (∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥) ↔ ∀𝑥(𝜑𝐴𝐸)))
98pm5.32i 564 . 2 ((𝐴𝐶 ∧ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥)) ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑𝐴𝐸)))
101, 4, 93bitri 286 1 (𝐴𝐵 ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑𝐴𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1629   ∈ wcel 2145  {cab 2757  ∩ cint 4611  ‘cfv 6031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-sn 4317  df-pr 4319  df-uni 4575  df-int 4612  df-iota 5994  df-fv 6039 This theorem is referenced by:  elcnvintab  38434
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