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Theorem elmapintab 43091
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 {𝑥𝜑} 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 6905 . . . 4 (𝐹𝐴) ∈ V
32elintab 4956 . . 3 ((𝐹𝐴) ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥))
43anbi2i 621 . 2 ((𝐴𝐶 ∧ (𝐹𝐴) ∈ {𝑥𝜑}) ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥)))
5 elmapintab.2 . . . . . 6 (𝐴𝐸 ↔ (𝐴𝐶 ∧ (𝐹𝐴) ∈ 𝑥))
65baibr 535 . . . . 5 (𝐴𝐶 → ((𝐹𝐴) ∈ 𝑥𝐴𝐸))
76imbi2d 339 . . . 4 (𝐴𝐶 → ((𝜑 → (𝐹𝐴) ∈ 𝑥) ↔ (𝜑𝐴𝐸)))
87albidv 1915 . . 3 (𝐴𝐶 → (∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥) ↔ ∀𝑥(𝜑𝐴𝐸)))
98pm5.32i 573 . 2 ((𝐴𝐶 ∧ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥)) ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑𝐴𝐸)))
101, 4, 93bitri 296 1 (𝐴𝐵 ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑𝐴𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wcel 2098  {cab 2702   cint 4944  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-nul 5301
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-sn 4625  df-pr 4627  df-uni 4904  df-int 4945  df-iota 6495  df-fv 6551
This theorem is referenced by:  elcnvintab  43097
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