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Theorem pgindnf 47926
Description: Version of pgind 47927 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 47920 . 2 Pg = setrecs((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)))
2 pgindnf.4 . 2 (𝑦 = 𝐴 → (𝜒𝜃))
3 pgindnf.1 . . . . . . 7 𝑥𝜑
4 nfv 1916 . . . . . . 7 𝑥𝑦𝑧 𝜒
53, 4nfan 1901 . . . . . 6 𝑥(𝜑 ∧ ∀𝑦𝑧 𝜒)
6 pgindlem 47925 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑧)
76sseld 3981 . . . . . . . . . 10 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥)) → 𝑦𝑧))
87imim1d 82 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((𝑦𝑧𝜒) → (𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥)) → 𝜒)))
98ralimdv2 3162 . . . . . . . 8 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → (∀𝑦𝑧 𝜒 → ∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒))
10 pgindnf.5 . . . . . . . . 9 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
111019.21bi 2181 . . . . . . . 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 3253 . . . . 5 ((𝜑 ∧ ∀𝑦𝑧 𝜒) → ∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓)
15 pgindnf.3 . . . . . 6 (𝑥 = 𝑦 → (𝜓𝜒))
16 vex 3477 . . . . . . . . 9 𝑧 ∈ V
17 pweq 4616 . . . . . . . . . . 11 (𝑎 = 𝑧 → 𝒫 𝑎 = 𝒫 𝑧)
1817sqxpeqd 5708 . . . . . . . . . 10 (𝑎 = 𝑧 → (𝒫 𝑎 × 𝒫 𝑎) = (𝒫 𝑧 × 𝒫 𝑧))
19 eqid 2731 . . . . . . . . . 10 (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)) = (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))
20 vpwex 5375 . . . . . . . . . . 11 𝒫 𝑧 ∈ V
2120, 20xpex 7744 . . . . . . . . . 10 (𝒫 𝑧 × 𝒫 𝑧) ∈ V
2218, 19, 21fvmpt 6998 . . . . . . . . 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 2740 . . . . . . 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 3942 . . . . 5 (∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓 ↔ ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒)
2714, 26sylib 217 . . . 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 1929 . 2 (𝜑 → ∀𝑧(∀𝑦𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
301, 2, 29setis 47908 1 (𝜑 → (𝐴 ∈ Pg → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1538   = wceq 1540  wnf 1784  wcel 2105  wral 3060  Vcvv 3473  cun 3946  𝒫 cpw 4602  cmpt 5231   × cxp 5674  cfv 6543  1st c1st 7977  2nd c2nd 7978  Pgcpg 47919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7979  df-2nd 7980  df-setrecs 47894  df-pg 47920
This theorem is referenced by:  pgind  47927
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