Users' Mathboxes Mathbox for Emmett Weisz < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pgindnf Structured version   Visualization version   GIF version

Theorem pgindnf 48947
Description: Version of pgind 48948 with extraneous not-free requirements. (Contributed by Emmett Weisz, 27-May-2024.) (New usage is discouraged.)
Hypotheses
Ref Expression
pgindnf.1 𝑥𝜑
pgindnf.2 𝑦𝜑
pgindnf.3 (𝑥 = 𝑦 → (𝜓𝜒))
pgindnf.4 (𝑦 = 𝐴 → (𝜒𝜃))
pgindnf.5 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
Assertion
Ref Expression
pgindnf (𝜑 → (𝐴 ∈ Pg → 𝜃))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝜒,𝑥   𝜓,𝑦   𝜃,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)   𝐴(𝑥)

Proof of Theorem pgindnf
Dummy variables 𝑎 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pg 48941 . 2 Pg = setrecs((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)))
2 pgindnf.4 . 2 (𝑦 = 𝐴 → (𝜒𝜃))
3 pgindnf.1 . . . . . . 7 𝑥𝜑
4 nfv 1912 . . . . . . 7 𝑥𝑦𝑧 𝜒
53, 4nfan 1897 . . . . . 6 𝑥(𝜑 ∧ ∀𝑦𝑧 𝜒)
6 pgindlem 48946 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑧)
76sseld 3994 . . . . . . . . . 10 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥)) → 𝑦𝑧))
87imim1d 82 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((𝑦𝑧𝜒) → (𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥)) → 𝜒)))
98ralimdv2 3161 . . . . . . . 8 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (∀𝑦𝑧 𝜒 → ∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒))
10 pgindnf.5 . . . . . . . . 9 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
111019.21bi 2187 . . . . . . . 8 (𝜑 → (∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
129, 11sylan9r 508 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)) → (∀𝑦𝑧 𝜒𝜓))
1312impancom 451 . . . . . 6 ((𝜑 ∧ ∀𝑦𝑧 𝜒) → (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → 𝜓))
145, 13ralrimi 3255 . . . . 5 ((𝜑 ∧ ∀𝑦𝑧 𝜒) → ∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓)
15 pgindnf.3 . . . . . 6 (𝑥 = 𝑦 → (𝜓𝜒))
16 vex 3482 . . . . . . . . 9 𝑧 ∈ V
17 pweq 4619 . . . . . . . . . . 11 (𝑎 = 𝑧 → 𝒫 𝑎 = 𝒫 𝑧)
1817sqxpeqd 5721 . . . . . . . . . 10 (𝑎 = 𝑧 → (𝒫 𝑎 × 𝒫 𝑎) = (𝒫 𝑧 × 𝒫 𝑧))
19 eqid 2735 . . . . . . . . . 10 (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)) = (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))
20 vpwex 5383 . . . . . . . . . . 11 𝒫 𝑧 ∈ V
2120, 20xpex 7772 . . . . . . . . . 10 (𝒫 𝑧 × 𝒫 𝑧) ∈ V
2218, 19, 21fvmpt 7016 . . . . . . . . 9 (𝑧 ∈ V → ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧) = (𝒫 𝑧 × 𝒫 𝑧))
2316, 22ax-mp 5 . . . . . . . 8 ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧) = (𝒫 𝑧 × 𝒫 𝑧)
2423eqcomi 2744 . . . . . . 7 (𝒫 𝑧 × 𝒫 𝑧) = ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)
2524a1i 11 . . . . . 6 (𝑥 = 𝑦 → (𝒫 𝑧 × 𝒫 𝑧) = ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧))
2615, 25cbvralv2 3957 . . . . 5 (∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓 ↔ ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒)
2714, 26sylib 218 . . . 4 ((𝜑 ∧ ∀𝑦𝑧 𝜒) → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒)
2827ex 412 . . 3 (𝜑 → (∀𝑦𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
2928alrimiv 1925 . 2 (𝜑 → ∀𝑧(∀𝑦𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
301, 2, 29setis 48929 1 (𝜑 → (𝐴 ∈ Pg → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wnf 1780  wcel 2106  wral 3059  Vcvv 3478  cun 3961  𝒫 cpw 4605  cmpt 5231   × cxp 5687  cfv 6563  1st c1st 8011  2nd c2nd 8012  Pgcpg 48940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014  df-setrecs 48915  df-pg 48941
This theorem is referenced by:  pgind  48948
  Copyright terms: Public domain W3C validator