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Theorem elmapintab 36818
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 5996 . . . 4 (𝐹𝐴) ∈ V
32elintab 4320 . . 3 ((𝐹𝐴) ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥))
43anbi2i 725 . 2 ((𝐴𝐶 ∧ (𝐹𝐴) ∈ {𝑥𝜑}) ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥)))
5 elmapintab.2 . . . . . 6 (𝐴𝐸 ↔ (𝐴𝐶 ∧ (𝐹𝐴) ∈ 𝑥))
65baibr 942 . . . . 5 (𝐴𝐶 → ((𝐹𝐴) ∈ 𝑥𝐴𝐸))
76imbi2d 328 . . . 4 (𝐴𝐶 → ((𝜑 → (𝐹𝐴) ∈ 𝑥) ↔ (𝜑𝐴𝐸)))
87albidv 1802 . . 3 (𝐴𝐶 → (∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥) ↔ ∀𝑥(𝜑𝐴𝐸)))
98pm5.32i 666 . 2 ((𝐴𝐶 ∧ ∀𝑥(𝜑 → (𝐹𝐴) ∈ 𝑥)) ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑𝐴𝐸)))
101, 4, 93bitri 284 1 (𝐴𝐵 ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑𝐴𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wcel 1938  {cab 2500   cint 4308  cfv 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-nul 4616
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-sn 4029  df-pr 4031  df-uni 4271  df-int 4309  df-iota 5653  df-fv 5697
This theorem is referenced by:  elcnvintab  36824
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