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Theorem elmapintab 37721
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 6188 . . . 4 (𝐹𝐴) ∈ V
32elintab 4478 . . 3 ((𝐹𝐴) ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥))
43anbi2i 729 . 2 ((𝐴𝐶 ∧ (𝐹𝐴) ∈ {𝑥𝜑}) ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥)))
5 elmapintab.2 . . . . . 6 (𝐴𝐸 ↔ (𝐴𝐶 ∧ (𝐹𝐴) ∈ 𝑥))
65baibr 944 . . . . 5 (𝐴𝐶 → ((𝐹𝐴) ∈ 𝑥𝐴𝐸))
76imbi2d 330 . . . 4 (𝐴𝐶 → ((𝜑 → (𝐹𝐴) ∈ 𝑥) ↔ (𝜑𝐴𝐸)))
87albidv 1847 . . 3 (𝐴𝐶 → (∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥) ↔ ∀𝑥(𝜑𝐴𝐸)))
98pm5.32i 668 . 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 384  wal 1479  wcel 1988  {cab 2606   cint 4466  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-sn 4169  df-pr 4171  df-uni 4428  df-int 4467  df-iota 5839  df-fv 5884
This theorem is referenced by:  elcnvintab  37727
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