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Theorem pgindnf 47473
Description: Version of pgind 47474 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 47467 . 2 Pg = setrecs((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)))
2 pgindnf.4 . 2 (𝑦 = 𝐴 → (𝜒𝜃))
3 pgindnf.1 . . . . . . 7 𝑥𝜑
4 nfv 1917 . . . . . . 7 𝑥𝑦𝑧 𝜒
53, 4nfan 1902 . . . . . 6 𝑥(𝜑 ∧ ∀𝑦𝑧 𝜒)
6 pgindlem 47472 . . . . . . . . . . 11 (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → ((1st𝑥) ∪ (2nd𝑥)) ⊆ 𝑧)
76sseld 3978 . . . . . . . . . 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 2182 . . . . . . . 8 (𝜑 → (∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
129, 11sylan9r 509 . . . . . . 7 ((𝜑𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)) → (∀𝑦𝑧 𝜒𝜓))
1312impancom 452 . . . . . 6 ((𝜑 ∧ ∀𝑦𝑧 𝜒) → (𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧) → 𝜓))
145, 13ralrimi 3254 . . . . 5 ((𝜑 ∧ ∀𝑦𝑧 𝜒) → ∀𝑥 ∈ (𝒫 𝑧 × 𝒫 𝑧)𝜓)
15 pgindnf.3 . . . . . 6 (𝑥 = 𝑦 → (𝜓𝜒))
16 vex 3478 . . . . . . . . 9 𝑧 ∈ V
17 pweq 4611 . . . . . . . . . . 11 (𝑎 = 𝑧 → 𝒫 𝑎 = 𝒫 𝑧)
1817sqxpeqd 5702 . . . . . . . . . 10 (𝑎 = 𝑧 → (𝒫 𝑎 × 𝒫 𝑎) = (𝒫 𝑧 × 𝒫 𝑧))
19 eqid 2732 . . . . . . . . . 10 (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎)) = (𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))
20 vpwex 5369 . . . . . . . . . . 11 𝒫 𝑧 ∈ V
2120, 20xpex 7724 . . . . . . . . . 10 (𝒫 𝑧 × 𝒫 𝑧) ∈ V
2218, 19, 21fvmpt 6985 . . . . . . . . 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 2741 . . . . . . 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 3939 . . . . 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 413 . . 3 (𝜑 → (∀𝑦𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
2928alrimiv 1930 . 2 (𝜑 → ∀𝑧(∀𝑦𝑧 𝜒 → ∀𝑦 ∈ ((𝑎 ∈ V ↦ (𝒫 𝑎 × 𝒫 𝑎))‘𝑧)𝜒))
301, 2, 29setis 47455 1 (𝜑 → (𝐴 ∈ Pg → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wnf 1785  wcel 2106  wral 3061  Vcvv 3474  cun 3943  𝒫 cpw 4597  cmpt 5225   × cxp 5668  cfv 6533  1st c1st 7957  2nd c2nd 7958  Pgcpg 47466
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fv 6541  df-1st 7959  df-2nd 7960  df-setrecs 47441  df-pg 47467
This theorem is referenced by:  pgind  47474
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