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Theorem elmapintrab 40319
 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 4852 . . 3 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶𝜑)} ↔ ∀𝑤 ∈ 𝒫 𝐵(∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)))
2 df-ral 3111 . . 3 (∀𝑤 ∈ 𝒫 𝐵(∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤) ↔ ∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)))
31, 2syl6bb 290 . 2 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶𝜑)} ↔ ∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤))))
4 velpw 4502 . . . . . 6 (𝑤 ∈ 𝒫 𝐵𝑤𝐵)
5 19.23v 1943 . . . . . . 7 (∀𝑥((𝑤 = 𝐶𝜑) → 𝐴𝑤) ↔ (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤))
65bicomi 227 . . . . . 6 ((∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤) ↔ ∀𝑥((𝑤 = 𝐶𝜑) → 𝐴𝑤))
74, 6imbi12i 354 . . . . 5 ((𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ (𝑤𝐵 → ∀𝑥((𝑤 = 𝐶𝜑) → 𝐴𝑤)))
8 19.21v 1940 . . . . 5 (∀𝑥(𝑤𝐵 → ((𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ (𝑤𝐵 → ∀𝑥((𝑤 = 𝐶𝜑) → 𝐴𝑤)))
9 bi2.04 392 . . . . . . 7 ((𝑤𝐵 → ((𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ ((𝑤 = 𝐶𝜑) → (𝑤𝐵𝐴𝑤)))
10 impexp 454 . . . . . . 7 (((𝑤 = 𝐶𝜑) → (𝑤𝐵𝐴𝑤)) ↔ (𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
119, 10bitri 278 . . . . . 6 ((𝑤𝐵 → ((𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ (𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
1211albii 1821 . . . . 5 (∀𝑥(𝑤𝐵 → ((𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ ∀𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
137, 8, 123bitr2i 302 . . . 4 ((𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ ∀𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
1413albii 1821 . . 3 (∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ ∀𝑤𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
15 alcom 2160 . . 3 (∀𝑤𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))) ↔ ∀𝑥𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))))
16 elmapintrab.ex . . . . . . 7 𝐶 ∈ V
17 sseq1 3940 . . . . . . . . 9 (𝑤 = 𝐶 → (𝑤𝐵𝐶𝐵))
18 eleq2 2878 . . . . . . . . . 10 (𝑤 = 𝐶 → (𝐴𝑤𝐴𝐶))
19 elmapintrab.sub . . . . . . . . . . . 12 𝐶𝐵
2019sseli 3911 . . . . . . . . . . 11 (𝐴𝐶𝐴𝐵)
2120pm4.71ri 564 . . . . . . . . . 10 (𝐴𝐶 ↔ (𝐴𝐵𝐴𝐶))
2218, 21syl6bb 290 . . . . . . . . 9 (𝑤 = 𝐶 → (𝐴𝑤 ↔ (𝐴𝐵𝐴𝐶)))
2317, 22imbi12d 348 . . . . . . . 8 (𝑤 = 𝐶 → ((𝑤𝐵𝐴𝑤) ↔ (𝐶𝐵 → (𝐴𝐵𝐴𝐶))))
2423imbi2d 344 . . . . . . 7 (𝑤 = 𝐶 → ((𝜑 → (𝑤𝐵𝐴𝑤)) ↔ (𝜑 → (𝐶𝐵 → (𝐴𝐵𝐴𝐶)))))
2516, 24ceqsalv 3479 . . . . . 6 (∀𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))) ↔ (𝜑 → (𝐶𝐵 → (𝐴𝐵𝐴𝐶))))
26 bi2.04 392 . . . . . 6 ((𝜑 → (𝐶𝐵 → (𝐴𝐵𝐴𝐶))) ↔ (𝐶𝐵 → (𝜑 → (𝐴𝐵𝐴𝐶))))
27 pm5.5 365 . . . . . . . 8 (𝐶𝐵 → ((𝐶𝐵 → (𝜑 → (𝐴𝐵𝐴𝐶))) ↔ (𝜑 → (𝐴𝐵𝐴𝐶))))
2819, 27ax-mp 5 . . . . . . 7 ((𝐶𝐵 → (𝜑 → (𝐴𝐵𝐴𝐶))) ↔ (𝜑 → (𝐴𝐵𝐴𝐶)))
29 jcab 521 . . . . . . 7 ((𝜑 → (𝐴𝐵𝐴𝐶)) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝐶)))
3028, 29bitri 278 . . . . . 6 ((𝐶𝐵 → (𝜑 → (𝐴𝐵𝐴𝐶))) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝐶)))
3125, 26, 303bitri 300 . . . . 5 (∀𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))) ↔ ((𝜑𝐴𝐵) ∧ (𝜑𝐴𝐶)))
3231albii 1821 . . . 4 (∀𝑥𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))) ↔ ∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝐶)))
33 19.26 1871 . . . 4 (∀𝑥((𝜑𝐴𝐵) ∧ (𝜑𝐴𝐶)) ↔ (∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶)))
34 19.23v 1943 . . . . 5 (∀𝑥(𝜑𝐴𝐵) ↔ (∃𝑥𝜑𝐴𝐵))
3534anbi1i 626 . . . 4 ((∀𝑥(𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶)))
3632, 33, 353bitri 300 . . 3 (∀𝑥𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤𝐵𝐴𝑤))) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶)))
3714, 15, 363bitri 300 . 2 (∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶𝜑) → 𝐴𝑤)) ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶)))
383, 37syl6bb 290 1 (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497  ∩ cint 4839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499  df-int 4840 This theorem is referenced by:  elinintrab  40320  cnvcnvintabd  40343  cnvintabd  40346
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