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Theorem elmapintrab 39820
Description: Two ways to say a set is an element of the intersection of a class of images. (Contributed by RP, 16-Aug-2020.)
Hypotheses
Ref Expression
elmapintrab.ex 𝐶 ∈ V
elmapintrab.sub 𝐶𝐵
Assertion
Ref Expression
elmapintrab (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶))))
Distinct variable groups:   𝜑,𝑤   𝑥,𝑤,𝐴   𝑤,𝐵,𝑥   𝑤,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝑉(𝑥,𝑤)

Proof of Theorem elmapintrab
StepHypRef Expression
1 elintrabg 4887 . . 3 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶𝜑)} ↔ ∀𝑤 ∈ 𝒫 𝐵(∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)))
2 df-ral 3148 . . 3 (∀𝑤 ∈ 𝒫 𝐵(∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤) ↔ ∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)))
31, 2syl6bb 288 . 2 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶𝜑)} ↔ ∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤))))
4 velpw 4550 . . . . . 6 (𝑤 ∈ 𝒫 𝐵𝑤𝐵)
5 19.23v 1936 . . . . . . 7 (∀𝑥((𝑤 = 𝐶𝜑) → 𝐴𝑤) ↔ (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤))
65bicomi 225 . . . . . 6 ((∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤) ↔ ∀𝑥((𝑤 = 𝐶𝜑) → 𝐴𝑤))
74, 6imbi12i 352 . . . . 5 ((𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ (𝑤𝐵 → ∀𝑥((𝑤 = 𝐶𝜑) → 𝐴𝑤)))
8 19.21v 1933 . . . . 5 (∀𝑥(𝑤𝐵 → ((𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ (𝑤𝐵 → ∀𝑥((𝑤 = 𝐶𝜑) → 𝐴𝑤)))
9 bi2.04 389 . . . . . . 7 ((𝑤𝐵 → ((𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ ((𝑤 = 𝐶𝜑) → (𝑤𝐵𝐴𝑤)))
10 impexp 451 . . . . . . 7 (((𝑤 = 𝐶𝜑) → (𝑤𝐵𝐴𝑤)) ↔ (𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
119, 10bitri 276 . . . . . 6 ((𝑤𝐵 → ((𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ (𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
1211albii 1813 . . . . 5 (∀𝑥(𝑤𝐵 → ((𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ ∀𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
137, 8, 123bitr2i 300 . . . 4 ((𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ ∀𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
1413albii 1813 . . 3 (∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ ∀𝑤𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
15 alcom 2155 . . 3 (∀𝑤𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))) ↔ ∀𝑥𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
16 elmapintrab.ex . . . . . . 7 𝐶 ∈ V
17 sseq1 3996 . . . . . . . . 9 (𝑤 = 𝐶 → (𝑤𝐵𝐶𝐵))
18 eleq2 2906 . . . . . . . . . 10 (𝑤 = 𝐶 → (𝐴𝑤𝐴𝐶))
19 elmapintrab.sub . . . . . . . . . . . 12 𝐶𝐵
2019sseli 3967 . . . . . . . . . . 11 (𝐴𝐶𝐴𝐵)
2120pm4.71ri 561 . . . . . . . . . 10 (𝐴𝐶 ↔ (𝐴𝐵𝐴𝐶))
2218, 21syl6bb 288 . . . . . . . . 9 (𝑤 = 𝐶 → (𝐴𝑤 ↔ (𝐴𝐵𝐴𝐶)))
2317, 22imbi12d 346 . . . . . . . 8 (𝑤 = 𝐶 → ((𝑤𝐵𝐴𝑤) ↔ (𝐶𝐵 → (𝐴𝐵𝐴𝐶))))
2423imbi2d 342 . . . . . . 7 (𝑤 = 𝐶 → ((𝜑 → (𝑤𝐵𝐴𝑤)) ↔ (𝜑 → (𝐶𝐵 → (𝐴𝐵𝐴𝐶)))))
2516, 24ceqsalv 3538 . . . . . 6 (∀𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))) ↔ (𝜑 → (𝐶𝐵 → (𝐴𝐵𝐴𝐶))))
26 bi2.04 389 . . . . . 6 ((𝜑 → (𝐶𝐵 → (𝐴𝐵𝐴𝐶))) ↔ (𝐶𝐵 → (𝜑 → (𝐴𝐵𝐴𝐶))))
27 pm5.5 363 . . . . . . . 8 (𝐶𝐵 → ((𝐶𝐵 → (𝜑 → (𝐴𝐵𝐴𝐶))) ↔ (𝜑 → (𝐴𝐵𝐴𝐶))))
2819, 27ax-mp 5 . . . . . . 7 ((𝐶𝐵 → (𝜑 → (𝐴𝐵𝐴𝐶))) ↔ (𝜑 → (𝐴𝐵𝐴𝐶)))
29 jcab 518 . . . . . . 7 ((𝜑 → (𝐴𝐵𝐴𝐶)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝐶)))
3028, 29bitri 276 . . . . . 6 ((𝐶𝐵 → (𝜑 → (𝐴𝐵𝐴𝐶))) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝐶)))
3125, 26, 303bitri 298 . . . . 5 (∀𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝐶)))
3231albii 1813 . . . 4 (∀𝑥𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))) ↔ ∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝐶)))
33 19.26 1864 . . . 4 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝐶)) ↔ (∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶)))
34 19.23v 1936 . . . . 5 (∀𝑥(𝜑𝐴𝐵) ↔ (∃𝑥𝜑𝐴𝐵))
3534anbi1i 623 . . . 4 ((∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶)))
3632, 33, 353bitri 298 . . 3 (∀𝑥𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶)))
3714, 15, 363bitri 298 . 2 (∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶)))
383, 37syl6bb 288 1 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528   = wceq 1530  wex 1773  wcel 2107  wral 3143  {crab 3147  Vcvv 3500  wss 3940  𝒫 cpw 4542   cint 4874
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rab 3152  df-v 3502  df-in 3947  df-ss 3956  df-pw 4544  df-int 4875
This theorem is referenced by:  elinintrab  39821  cnvcnvintabd  39844  cnvintabd  39847
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