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Theorem pgindnf 49235
Description: Version of pgind 49236 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 49229 . 2 Pg = setrecs((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)))
2 pgindnf.4 . 2 (𝑦 = 𝐴 → (𝜒𝜃))
3 pgindnf.1 . . . . . . 7 𝑥𝜑
4 nfv 1914 . . . . . . 7 𝑥𝑦𝑧 𝜒
53, 4nfan 1899 . . . . . 6 𝑥(𝜑 ∧ ∀𝑦𝑧 𝜒)
6 pgindlem 49234 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑧)
76sseld 3982 . . . . . . . . . 10 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥)) → 𝑦𝑧))
87imim1d 82 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((𝑦𝑧𝜒) → (𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥)) → 𝜒)))
98ralimdv2 3163 . . . . . . . 8 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (∀𝑦𝑧 𝜒 → ∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒))
10 pgindnf.5 . . . . . . . . 9 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
111019.21bi 2189 . . . . . . . 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 3257 . . . . 5 ((𝜑 ∧ ∀𝑦𝑧 𝜒) → ∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓)
15 pgindnf.3 . . . . . 6 (𝑥 = 𝑦 → (𝜓𝜒))
16 vex 3484 . . . . . . . . 9 𝑧 ∈ V
17 pweq 4614 . . . . . . . . . . 11 (𝑎 = 𝑧 → 𝒫 𝑎 = 𝒫 𝑧)
1817sqxpeqd 5717 . . . . . . . . . 10 (𝑎 = 𝑧 → (𝒫 𝑎 × 𝒫 𝑎) = (𝒫 𝑧 × 𝒫 𝑧))
19 eqid 2737 . . . . . . . . . 10 (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)) = (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))
20 vpwex 5377 . . . . . . . . . . 11 𝒫 𝑧 ∈ V
2120, 20xpex 7773 . . . . . . . . . 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 2746 . . . . . . 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 3945 . . . . 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 1927 . 2 (𝜑 → ∀𝑧(∀𝑦𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
301, 2, 29setis 49217 1 (𝜑 → (𝐴 ∈ Pg → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wnf 1783  wcel 2108  wral 3061  Vcvv 3480  cun 3949  𝒫 cpw 4600  cmpt 5225   × cxp 5683  cfv 6561  1st c1st 8012  2nd c2nd 8013  Pgcpg 49228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015  df-setrecs 49203  df-pg 49229
This theorem is referenced by:  pgind  49236
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